\section{Introduction and statements}
 
 
The main result of the present paper is the construction of a natural transformation 
\begin{equation}\label{wefewrfwewefwerfwef}
K^{G,\cX}_{\bC}\to K^{G,\mathrm{An}}_{\bC}
\end{equation} 
between two versions of spectrum-valued equivariant {$K$-homology} functors {which is} an equivalence when evaluated on proper and $G$-finite simplicial complexes,  {where $G$ is a  {countable} discrete  group}. Following classical terminology, we call this transformation the {Paschke transformation}.
Here $G$ is a discrete group, the functor  $K^{G,\cX}_{\bC}$ in the domain is derived from a coarse $K$-theory functor using {coarse geometric constructions},
while the target $ K^{G,\mathrm{An}}_{\bC}$ is a spectrum-valued version of the classical equivariant analytic $K$-homology. {In both versions, }the subscript $\bC$ indicates  the  dependence of these functors  on the choice of a coefficient $G$-$C^{*}$-category $\bC$, {or more precisely, on a coefficient pair $(\bC,{\mathbf{K}})$ consisting of a unital $G$-$C^*$-category $\bC$ together with an invariant ideal ${\mathbf{K}}$. Often, the ideal ${\mathbf{K}}$ will be clear from the choice of the unital $G$-$C^*$-category $\bC$, we therefore refrain from adding it to the notation}. 


The second main result of the present paper is Theorem \ref{wtoiguwegwergergregwe}, {which provides a direct} comparison of a Davis--L\"uck type assembly map with the Baum--Connes assembly map.
 
{In the following} we give a slightly informal description of our main result. Then we explain some relations to the classical Paschke duality, state the precise version our main result as Theorem \ref{qreoigjoergegqrgqerqfewf}, {and finally discuss the comparison of assembly maps.}


{We emphasize that this paper is not the first to treat the topic of equivariant Paschke duality and comparisons of assembly maps, most current are the papers \cite{Benameur:2020aa} and \cite{kranz}. We explain more about this in Remarks~\ref{Remark:previous-work} and \ref{Remark:comparison-assembly}.}


\subsubsection*{The coefficient pair}
Both $K$-homology  theory functors  occuring in \eqref{wefewrfwewefwerfwef} 
depend on the choice of a  pair $( \bC , {\mathbf{K}} )$ of a unital $G$-$C^{*}$-category $\bC$ with an invariant ideal ${\mathbf{K}}$. This pair  gives rise to {the {Calkin}} $G$-$C^{*}$-category $\mathbf{Q}^{(G)}_{\mathrm{std}}$, 
see  Definition~\ref{qeroighjoreqgreqfefwfqwef} and \eqref{wefwfwfwefflkjlqwepofp}  {for the details}. 


The following example
  will be referred to as the case of {trivial coefficients}.
In this  case   $\bC $  is the $C^{*}$-category $\mathbf{Hilb}^{G}(\C)$ of pairs $(H,\rho)$ of Hilbert spaces with a unitary  $G$-action and bounded (not necessarily equivariant) operators with induced $G$-action by conjugation. The ideal      ${\mathbf{K}}$ is the wide subcategory $ \mathbf{Hilb}_{c}^{G}(\C)$ of compact operators. By specializing the notation introduced in  Definition~\ref{qeroighjoreqgreqfefwfqwef} we let $\mathbf{Hilb}^{G}(\C)_{\mathrm{std}}$ be the full subcategory of   $\mathbf{Hilb}^{G}(\C)$ of  those pairs $(H,\rho)$
which are isomorphic to $(L^{2}(G)\otimes H' ,\lambda\otimes \id_{H'})$, where $\lambda$ is the left-regular representation and $H$ is some auxiliary Hilbert space. 
It contains the ideal   $\mathbf{Hilb}_{c}^{G}(\C)_{\mathrm{std}}\coloneqq\mathbf{Hilb}_{c}^{G}(\C)\cap  \mathbf{Hilb}^{G}(\C)_{\mathrm{std}}$. 
 Then  $\mathbf{Q}^{(G)}_{\mathrm{std}}$ is the Calkin category $\mathbf{Hilb}^{G}(\C)_{\mathrm{std}}/ \mathbf{Hilb}_{c}^{G}(\C)_{\mathrm{std}}$. Its objects are the $G$-Hilbert spaces  $(H,\rho)$ in $\mathbf{Hilb}^{G}(\C)_{\mathrm{std}}$, and its morphism spaces  are the quotient spaces  of bounded operators by compact operators with the induced $G$-action. In particular, the endomorphism algebra of each object $(H,\rho)$ is the usual Calkin algebra $Q(H)$ of $H$ with the $G$-action by conjugation, hence the name.

More generally, for a $G$-$C^*$-algebra $A$ we consider the pair $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^G(A),\mathbf{Hilb}^G_c(A))$ of Hilbert-$A$-modules with bounded and compact operators, respectively, see \cite{cank} and Example~\ref{weriogwergw} below. 
 Recall from \cite[{Def.\ 1.10}]{KKG} that there is a canonical functor $${\mathrm{kk}^{G}} \colon \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}}) \to \mathrm{KK}^{G}\, ,$$ where $\mathrm{KK}^{G}$ is a stable $\infty$-category refining the classical triangulated category for equivariant KK-theory of \cite{MR2193334}. If $A$ is unital, then we can compare $A$ and  the Calkin category   $\mathbf{Q}_{\mathrm{std}}^{ {(}G {)}}$ via
  a canonical map $\mathrm{kk}^{G}(\mathbf{Q}^{(G)}_\mathrm{std}) \to \Sigma \mathrm{kk}^{G}(A)$ in $\mathrm{KK}^{G}$. {This map}  induces an equivalence on 
  $\mathrm{KK}^{G}(P,-)$ for suitably finite objects $P$ {of $\mathrm{KK}^{G}$}, e.g.\  {the commutative $G$-$C^{*}$-algebras $C_0(X)$} for {$G$-finite} $G$-CW-complexes $X$. We refer to the {proof of Proposition \ref{wegojeogrregregwergwegre}} for details.
 
 
 
\subsubsection*{Analytic $K$-homology}
In order to define the equivariant  {analytic} $K$-homology functor $K^{G,\mathrm{An}}_{\bC}$ with coefficients in $(\bC,{\mathbf{K}})$, we use the bivariant spectrum-valued {refinement}  
\[\mathrm{KK}^{G}(-,-) \colon \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})^{\mathrm{op}}\times \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\to \Sp\]
of Kasparov's $K\!K^{G}_{*}$-theory {and its extension to $G$-$C^*$-categories} constructed in \cite{KKG} {and mentioned in the previous paragraph}.
The functor $K^{G,\mathrm{An}}_\bC$ is then defined by the formula
\begin{equation}\label{vfdsvsvvsvsdvdsadsvdsva}
K^{G,\mathrm{An}}_{\bC} \colon G\mathbf{LCH}^{\mathrm{prop}}_{+} \to \Sp^{\mathrm{}}\, , \quad X\mapsto \mathrm{KK}^{G}(C_{0}(X),\mathbf{Q}_{\mathrm{std}}^{(G)})\, .
\end{equation} 
The domain of this functor is the category $G\mathbf{LCH}^{\mathrm{prop}}_{+}$  of  locally compact {Hausdorff} $G$-spaces with partially defined and proper maps. {Equivalently, $G\mathbf{LCH}^{\mathrm{prop}}_{+}$ is the} Gelfand dual of the category of non-unital  commutative $G$-$C^{*}$-algebras. 
 
{The connection with the notation from \cite[Def.\ 1.13]{KKG} is given by
$$K^{G,\mathrm{An}}_{\bC}=K^{G,\mathrm{an}}_{\mathbf{Q}^{(G)}_{\mathrm{std}}}\ .$$ 
 {In particular, $K^{G,\mathrm{An}}_\bC$ is different from $K^{G,\mathrm{an}}_\bC$ --  we apologize for this notational incovenience.}
} 


We refer to \cite[Thm.\ 1.14]{KKG} for a list of the basic properties of $K^{G,\mathrm{An}}_{\bC}$.
In particular, $K^{G,\mathrm{An}}_{\bC}$ is homotopy invariant, excisive for closed decompositions of second countable spaces, and annihilates spaces of the form $[0,\infty)\times X$.


{For a $G$-space $X$ which is homotopy equivalent to a $G$-finite CW-complex with finite stabilizers and trivial coefficients, we have a natural isomorphism}
\begin{equation}\label{qefqwefwefewdq}
\pi_{*}K^{G,\mathrm{An}}_{\bC}(X)\cong K\!K^{G}_{*-1}(C_{0}(X),\C) \, .
\end{equation}
{see Proposition \ref{wegojeogrregregwergwegre}.}
This isomorphism {identifies our definition of equivariant {analytic}  $K$-homology with the classical definition {given} by 
 the right hand side of \eqref{qefqwefwefewdq}, up to a shift of degrees.}
 

In order to deal correctly with non-$G$-compact spaces in $G\mathbf{LCH}^{\mathrm{prop}}_{+}$ we will   consider
the  {locally finite} version $K^{G,\mathrm{An},\mathrm{lf}}_{\bC}$ of $K^{G,\mathrm{An}}_{\bC}$ which is defined as follows. If $X$ is in $ G\mathbf{LCH}^{\mathrm{prop}}_{+}$ and $U$ is an open $G$-invariant subset  of $X$ with $G$-compact closure, then we have a morphism
$X\to U$ in $ G\mathbf{LCH}^{\mathrm{prop}}_{+}$ given by the partially defined map
$X\supset U\stackrel{\id_{U}}{\to} U$   {which corresponds to} the extension-by-zero homomorphism $C_{0}(U)\to C_{0}(X)$ {on the level} of $G$-$C^{*}$-algebras. We define\begin{equation}\label{wergeefeferfewrferf}
K^{G,\mathrm{An},\mathrm{lf}}_{\bC}(X)\coloneqq\lim_{U\subseteq X}   K^{G,\mathrm{An}}_{\bC}(U)\ ,
\end{equation}  where the limit runs over all open subsets $U$ of $X$ with $G$-compact closure.
Using right Kan extensions,
one can turn this prescription into the definition of a functor

\begin{equation}\label{regrwefefefewf}
K^{G,\mathrm{An},\mathrm{lf}}_{\bC}\colon G\mathbf{LCH}^{\mathrm{prop}}_{+}\to \Sp\ ,
\end{equation} 
see \cite[Sec. 7.1.2]{buen} for a similar construction.
We have a natural transformation \begin{equation}\label{qwefwefwqoiudjoqwdwedqwedweqdewd}
c:K^{G,\mathrm{An}}_{\bC}\to K^{G,\mathrm{An},\mathrm{lf}}_{\bC}
\end{equation}
of functors from $G\mathbf{LCH}^{\mathrm{prop}}_{+}$ to $\Sp$.
 The functor $K^{G,\mathrm{An},\mathrm{lf}}_{\bC}$ is homotopy invariant. Its restriction 
to second countable spaces with proper $G$-action  is excisive for closed decompositions by \cite[Prop.\ 1.12.1]{KKG}. Finally, it sends countable disjoint unions to products.
If $X$ is $G$-compact, then the canonical map
$c_{X}:K^{G,\mathrm{An}}_{\bC}(X)\to K^{G,\mathrm{An},\mathrm{lf}}_{\bC}(X)$ is an equivalence. We refer to 
Proposition \ref{werigowergrefwerfwe} for a calculation of the  values of $K^{G,\mathrm{An},\mathrm{lf}}_{\bC}$ on more general spaces.


\subsubsection*{Coarse $K$-homology}
{
We now turn to a brief description of the functor $K^{G,\cX}_{\bC}$. More details will be given later.
For our purposes, the functor $K^{G,\cX}_{\bC}$ is most naturally defined on {the category $G\mathbf{UBC}$ of} $G$-uniform bornological coarse spaces.
{This category comes with a} cone-at-$\infty$ functor ${\mathcal{O}}^\infty \colon G\mathbf{UBC} \to G\mathbf{BC}$, {where $G\mathbf{BC}$ denotes the category of $G$-bornological coarse spaces}. Now, associated to the coefficient pair $(\bC,{\mathbf{K}})$, there is an equivariant coarse $K$-homology functor 
$K\bC\cX^G_{c,G_{can,max}}\colon G\mathbf{BC} \to \Sp$ {constructed in \cite{coarsek}}, and we set }
\begin{equation}\label{qewfoijfoiqwefqewfewfqewfd}
K^{G,\cX}_{\bC} \coloneqq K\bC\cX^G_{c,G_{can,max}}\circ {\mathcal{O}}^\infty \colon G\mathbf{UBC} \to \Sp\,.
\end{equation}
This composition is an equivariant local homology theory, i.e.\ it is homotopy invariant, excisive for closed decompositions, and vanishes on spaces of the form $[0,\infty) \otimes X$, see Prop. \ref{qroigjqowfewfqwefqw}.

\subsubsection*{A common domain for $K^{G,\mathrm{An}}_\bC$ and $K^{G,\cX}_\bC$}
By now, the functors $K^{G,\mathrm{An}}_{\bC}$ and $K^{G,\cX}_\bC$ cannot be compared. They are invariants of different objects: $G$-locally compact Hausdorff spaces on the one hand{,} and $G$-uniform bornological coarse spaces on the other hand. In Definition~\ref{qreoighwoefewqffqewffewfq} we introduce the category $G\mathbf{UBC}^\mathrm{scl}$ of  scaled $G$-uniform bornological coarse spaces  which  by definition,
comes with forgetful functors
$\iota^\mathrm{scl}$ to $G$-uniform bornological coarse spaces and $\iota^\mathrm{top}$ to $G$-locally compact Hausdorff spaces. Equipping a $G$-simplicial complex with the spherical metric on its simplices and a canonical  scale 
induces the upper part of the following diagram, where $G\mathbf{Simpl}_{\mathrm{fin}}^\mathrm{prop}$ denotes the category  of $G$-finite simplicial complexes with finite stabilizers and equivariant proper simplicial maps. 



We can summarize our first main result, slightly informally, by the following diagram.
\[\begin{tikzcd}
	& G\mathbf{Simpl}_{\mathrm{fin}}^{\mathrm{prop}} \ar[d] \ar[dl] \ar[dr] & \\
	G\mathbf{UBC} \ar[ddr,"{K^{G,\cX}_\bC}"', bend right]& G\mathbf{UBC}^\mathrm{scl} \ar[l,"\iota^\mathrm{scl}"'] \ar[r,"\iota^\mathrm{top}"] & G\mathbf{LCH}^{\mathrm{prop}}_{+} \ar[ddl,"K^{G,\mathrm{An}}_{\bC}", bend left] \\
	 & \stackrel{p}{\Longrightarrow} & \\
	 & \Sp &
\end{tikzcd}\]
The Paschke transformation $p$ will be constructed as a natural transformation between the two composites of the lower square. Equivalently, naturality  of $p$ can be stated by saying that it makes the lower square {lax-commutative}. We then show that the Paschke transformation renders 
 the large square commutative. In other words, the Paschke transformation becomes a natural equivalence when 
 restricted to $G$-finite and $G$-proper simplicial complexes. We will state our main theorem more formally as Theorem~\ref{qreoigjoergegqrgqerqfewf} below.


\subsubsection*{A review of classical Paschke duality}


{In order to motivate the definitions involved in the above diagram, we now review some aspects of classical Paschke duality.}
Based on the seminal work of Paschke  \cite{paschke}, the general theme of Paschke duality is to express the {analytic}
  $K$-homology   $$K_{*}^{\mathrm{an}}(X)\coloneqq KK_{*}(C_{0}(X),\C)$$ in terms of the $K$-theory
  of a $C^{*}$-algebra naturally associated to $X$, which is then often referred to as the {Paschke dual algebra} of $X$.

Classically, this is implemented as follows.
Let $X$ be a proper metric space and $\phi \colon C_{0}(X)\to B(H)$ be  a homomorphism of $C^{*}$-algebras, where  $H$ is a separable Hilbert space. To this data one associates  an exact sequence of $C^{*}$-algebras 
\begin{equation}\label{wergpowkgpowergregregwegrg}
0\to C(H,\phi)\to D( H,\phi)\to Q (H,\phi)\to 0
\end{equation}
where $D(H,\phi)$ is the $C^*$-subalgebra of $B(H)$ generated by the controlled and pseudolocal operators and $C(H,\phi)$,  called the {Roe algebra}, is its ideal generated by the operators which are in addition locally compact.

{If  $(H,\phi)$  is 
 sufficiently large ({very ample 
 in classical terminology or} 
 absorbing in the sense of Definition \ref{werogihetgergfdsf}),  then the $K$-theory of $Q(H,\phi)$ is a well-behaved invariant of $X$. More precisely, for a proper map $f \colon X \to X'$ and absorbing representations $(H,\phi)$ and $(H',\phi')$ for $X$ and $X'$ respectively, there exists a unitary, controlled and pseudolocal {isometry} $(H',\phi')\cong (H,\phi\circ f^{*})$ {called a {covering}, which is} unique up to conjugation by unitaries. This covering induces  a homomorphism  $D(H,\phi)\to D(H',\phi')$ preserving the respective Roe-algebras and therefore a homomorphism
 $Q(H,\phi)\to Q(H',\phi')$ between the quotients. For $f=\id_X$, this shows that the $K$-theory of $Q(H,\phi)$ is independent of the choice of an absorbing representation $(H,\phi)$. {We recall here that Voiculescu's Theorem grants the existence of such absorbing representations.}  {Furthermore,}  setting }
\[K_{*}^{\cX}(X) \coloneqq K^{C^{*}\mathrm{Alg}}_{*}(Q(H,\phi))\, .\]
{for {any choice of an} absorbing representation $(H,\phi)$, one obtains a functor}
\[K_{*}^{\cX}(-) \colon \Met^{\mathrm{prop}}\to \Ab^{\Z}\]
defined on the category of proper metric spaces and proper maps and taking values in graded abelian groups.  The superscript $\cX$  indicates the coarse geometric origin of the construction, whose implementation was
 initiated by Roe \cite{roe_exotic_cohomology}.
The functor $K_*^{\cX}(-)$ is homotopy invariant, admits Mayer--Vietoris sequences. In addition, there is a natural  Paschke duality isomorphism 
\begin{equation}\label{qwefqoihfoiqwefewfqwefef}
K_*^{\cX}(X) \cong K^\mathrm{an}_{*-1}(X)
\end{equation}
given by a concrete cycle level construction, see \cite{higson_roe} for details. 
 
  
 
\subsubsection*{The Paschke morphism following Quiao--Roe}


The paper \cite{quro} discusses a systematic approach to the isomorphism \eqref{qwefqoihfoiqwefewfqwefef}, whose basic idea we will now adapt to the equivariant situation. We continue to assume that the $G$-space $X$ is equipped with an absorbing representation $\phi\colon C_0(X) \to B(H,\rho)$ where $H$ is a separable Hilbert space equipped with a unitary $G$-action $\rho$.
The idea is to derive the isomorphism in \eqref{qwefqoihfoiqwefewfqwefef} from 
a multiplication map
\begin{equation}\label{efqwefewdq}
\mu_{X} \colon C_{0}(X)\otimes Q^{G}(X)\to Q(H ) \, ,
\end{equation}
 $$Q^{G}(X) \coloneqq Q^{G}(H,\rho,\phi) \coloneqq D^{G}(H,\rho,\phi)/C^{G}(H,\rho,\phi)\, ,$$
 where $D^{G}(H,\rho,\phi)$  and $C^{G}(H,\rho,\phi)$ are defined as in the non-equivariant case by just adding the condition that the controlled generators are {$G$}-invariant. Furthermore 
  $Q(H) = Q(H,\rho)$ is the Calkin algebra of $(H,\rho)$ with the induced $G$-action.
Using the multiplication map \eqref{efqwefewdq}, one may define a Paschke morphism as the composition 
\begin{equation}\label{rfqwpofkjpqowefewfeqwfqef}
 p^{(H,\rho,\phi)}_{X} \colon \mathrm{KK} (\C,Q^{G}(X))\xrightarrow{\delta_{X}} \mathrm{KK}^{G}  (C_{0}(X),C_{0}(X)\otimes Q^{G}(X))\xrightarrow{\mu_{X}} \mathrm{KK}^{G} (C_{0}(X),Q(H) )  \, .
\end{equation}
The map  $\delta_{X} \coloneqq C_{0}(X)\otimes -$ is the exterior product in equivariant KK-theory and is called the diagonal morphism. We note that the algebras $Q^{G}(X )$ and $Q (H)$ are not separable, which is why $E$-theory instead of $K\!K$-theory is used in \cite{quro}. However, the equivariant KK-theory of \cite{KKG} is well-defined for all $G$-$C^{*}$-algebras, so we can safely work with this version rather than with $E$-theory. 

With this more abstract definition, how can one show that the Paschke morphism induces an isomorphism on $K$-groups, at least for suitable spaces $X$?  Our strategy to answer this question is as follows. 
Suppose one could show that the maps $p^{(H,\rho,\phi)}_{X}$ in \eqref{rfqwpofkjpqowefewfeqwfqef} were the components of a natural transformation of functors with values in the $\infty$-category of spectra,
and that both the domain and target of the Paschke morphism are homotopy invariant and excisive\footnote{This is the spectrum analogue of the property of admitting Mayer--Vietoris sequences  for group-valued functors} as functors in $X$.
Then for {$G$}-finite $G$-CW-complexes $X$, by induction over the number of $G$-cells, one can reduce the verification  that $p_X^{(H,\rho,\phi)}$ is an equivalence 
to the  cases of $G$-orbits, i.e.\ spaces of the form $G/H$, where $H$ runs over the  subgroups of $G$ appearing as   stabilizer of the $G$-action on $X$. {While in the non-equivariant case, only the trivial case $X=\ast$ is to be treated, the verification that the Paschke maps are equivalences on general $G$-orbits is a non-trivial matter.}

{The above strategy} will indeed be the essential idea of the proof of our main Theorem \ref{qreoigjoergegqrgqerqfewf} below. The first difficulty to overcome is to show that the Paschke maps $p^{(H,\rho,\phi)}_{X}$ are indeed the components of a natural transformation, in particular, to show that the spectrum $\mathrm{KK}(\C,Q^{G}(X))$ appearing in the domain of the Paschke map, is a homotopy invariant and excisive functor in $X$ (it is not even a functor in any obvious manner).
The origin of the problem  is that in order to define $Q^{G}(X)=Q^{G}(H,\rho,\phi)$, one has to \emph{choose} an absorbing representation $(H,\rho,\phi)$, and for a morphism $X\to X'$ one  has to \emph{choose}  a covering in order to define the map 
$\mathrm{KK} (\C,Q^{G}(X))\to \mathrm{KK} (\C,Q^{G}(X'))$. 
Defined in this way, the resulting  map of  spectra depends on these choices and is, at best, unique up to an unspecified homotopy, which is not sufficient for our purposes.

\subsubsection*{The Paschke morphism in our setup}

Our key idea to overcome these functoriality issues is to work with the category of all representations.  In fact, the categories  of such representations themselves depend on the space in a strictly functorial manner. Their use hence circumvents the need to find absorbing representations.
The idea  to work with the whole category of representations is not new; it has first been exploited in \cite{buen} in order to define a spectrum-valued coarse $K$-homology functor $K\!\cX$.

In the present paper, as indicated earlier, we work with its equivariant generalization,  the equivariant  coarse $K$-homology functor 
\[K\bC\cX^{G}_{c} \colon G\mathbf{BC}\to \Sp\] 
introduced in \cite{coarsek}{, where, as before, $G\mathbf{BC}$ denotes the category of $G$-bornological coarse spaces}.
Again, the symbol $\bC$ refers to its dependence on a coefficient pair $(\bC,{\mathbf{K}})$.
In the case of trivial coefficients it is shown in  \cite[Thm.\ 6.1]{indexclass} that this functor is equivalent to the classical definition of equivariant coarse $K$-homology in terms of Roe algebras. More precisely, if   the    $G$-space $X$ is nice, and $C^G(X) \coloneqq C^G(H,\rho,\phi)$ with $(H,\rho,\phi)$ ample,
we have a natural equivalence
\[ K\bC\cX^{G}_{c}(X) \simeq K^{C^{*}\mathrm{Alg}}(C^{G}(X)) \, .\]


By construction, see Definition~\ref{qrogijeqoifefewfefewqffe}, for $X$ in $G\mathbf{BC}$  we have $K\bC\cX^{G}_{c}(X)=\mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)) $, where  
$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ is a $C^{*}$-category  of  equivariant $X$-controlled objects in $(\bC,{\mathbf{K}})$, see Definition~\ref{rfquhwfiuqwhfiufewqefqwefqwefwefqwef} for the details.
The endomorphism algebras of the objects of $  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$  are natural analogues
of the Roe algebras $C(H,\rho,\phi)$.

We now indicate the relation between the functor $X \mapsto K^{G,\cX}_\bC(X)$ and the association $X \mapsto \mathrm{KK}(\C,Q^G(X))$ appearing in the source of the Paschke morphism \eqref{rfqwpofkjpqowefewfeqwfqef}. Recall from \eqref{qewfoijfoiqwefqewfewfqewfd} that $K^{G,\cX}_\bC$ is defined as a composition of $K\bC\cX^{G}_{c}$ with the functor ${\mathcal{O}}^\infty(-)\otimes G_{can,max}$ on $G$-uniform bornological coarse spaces. 
 
 
 If $X$ is in $G\mathbf{UBC}$, then the cone ${\mathcal{O}}^{\infty}(X)$ is the $G$-set
 $\R\times X$ with a certain $G$-bornological coarse structure described in \cite[Sec.\ 9]{equicoarse} and \cite[Section 8]{ass}. 
 It contains the underlying $G$-bornological coarse space of $X$ as the subspace $\{0\}\times X$. We further let
 ${\mathcal{O}}(X)$ in $G\mathbf{BC}$ be the subset $[0,\infty)\times X$ with the induced structures.
The inclusion $X\to {\mathcal{O}}(X)$ induces an inclusion of categories \begin{equation}\label{r3foifjowqefewqfqewfef}
 \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,max})\to  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes G_{can,max})\, .
\end{equation}
to be thought of as the analog of the inclusion $C^G(X) \to D^G(X)$ in the classical situation, see Section~ \ref{weogiwjiergreggwrgwerg} for more details.
  The resulting quotient $C^{*}$-category  $\mathbf{Q}^{G}_{\bC}(X)$
 is then our version of the algebra $Q^{G}(X)$, and we have natural equivalence
\begin{equation}\label{gepojgregergegwgwergrweg}
 K^{G,\cX}_{\bC}(X)\simeq \mathrm{KK}(\C, \mathbf{Q}^{G}_{\bC}(X))\, .
\end{equation}   
   We refer to Section  \ref{qriugfhiergfedfqewfqewf9} and Lemma \ref{wrtoihgjoergergegwgrgwerg} for more details and necessary additions.  In particular we add scales $\tau$ (see Def. \ref{wfiuhiufqwefeewqfqwfewfqewf}) to the picture which leads to the better behaved version $\mathbf{Q}^{G}_{\bC}(X,\tau)$ of $\mathbf{Q}^{G}_{\bC}(X)$ which on the one hand satisfies  
   $\mathrm{KK}(\C,\mathbf{Q}^{G}_{\bC}(X,\tau))\simeq \mathrm{KK}(\C,\mathbf{Q}^{G}_{\bC}(X))$, but on the other hand allows the construction of a
 multiplication map
 \[ \mu_{(X,\tau)} \colon C_0(X) \otimes \mathbf{Q}^{G}_{\bC}(X,\tau) \to \mathbf{Q}^{(G)}_{\mathrm{std}},\]
see \eqref{oreihoijfvoisfdvervfdsvdfvsvv}, {where $\mathbf{Q}^{G}_{\bC}(X,\tau)$ is abbreviated to $\mathbf{Q}_\tau(X)$.}
In complete analogy to the earlier described Paschke morphism \eqref{rfqwpofkjpqowefewfeqwfqef}, we define our version of the Paschke morphism as the composition:
\begin{equation}\label{Paschke-morphisms}
	p_{(X,\tau)} \colon \mathrm{KK}(\C,\mathbf{Q}^{G}_{\bC}(X,\tau)) \stackrel{\delta_X}{\longrightarrow} \mathrm{KK}^{G}(C_0(X),C_0(X)\otimes \mathbf{Q}^{G}_{\bC}(X,\tau)) \stackrel{\mu_{(X,\tau)}}{\longrightarrow} \mathrm{KK}^{G}(C_0(X),\mathbf{Q}^{(G)}_{\mathrm{std}})\,.
\end{equation}
  
 
 The main result of this paper is then the following theorem. 

 \begin{theorem}\label{qreoigjoergegqrgqerqfewf}\mbox{}\begin{enumerate}
 \item\label{weiufhiqwefewwfqewfewf}
The morphisms in \eqref{Paschke-morphisms} assemble into a natural transformation of  spectrum-valued functors on $G\mathbf{UBC}^\mathrm{scl}$
\begin{equation}\label{qwoeifjoifjewfqwefwfqewfq}
p \colon K_{\bC}^{G,\cX}\circ \iota^{\mathrm{scl}}\to K_{\bC}^{G,\mathrm{An}}\circ \iota^{\mathrm{top}}\, .
\end{equation}
\item\label{qregiojqwfewfqwfqewf} If 
$(X,\tau)$ is in $G\mathbf{UBC}^{\mathrm{scl}}$ and homotopy equivalent to a $G$-finite
  $G$-simplicial complex with finite stabilizers, and 
the ideal ${\mathbf{K}}$ is compatible with orthogonal sums, then 
 $$p_{(X,\tau)} \colon K_{\bC}^{G,\cX}(X)\to K_{\bC}^{G,\mathrm{An}}(X)$$
 is an equivalence.
 \item \label{wrtigjiogowrefwerfewrf}
 If   $G$ is finite, 
$(X,\tau)$ is in $G\mathbf{UBC}^{\mathrm{scl}}$ and homotopy equivalent to a countable finite-dimensional
  $G$-simplicial complex, and 
the ideal ${\mathbf{K}}$ is compatible with orthogonal sums, then 
 $$  p^{\mathrm{lf}}_{(X,\tau)} \colon K_{\bC}^{G,\cX}(X)\to K_{\bC}^{G,\mathrm{An},{\mathrm{lf}}}(X)$$
 is an equivalence.
\end{enumerate}
 \end{theorem} We  refer to Definition~\ref{wefiqrjhfoifewqdqwedqwedqewd} for the condition  on ${\mathbf{K}}$ appearing in \ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf}.
 {In Assertion \ref{qreoigjoergegqrgqerqfewf}.\ref{wrtigjiogowrefwerfewrf}  we use the transformation  $c:K^{G,\mathrm{An}}_{\bC}\to K^{G,\mathrm{An},\mathrm{lf}}_{\bC}$ from \eqref{qwefwefwqoiudjoqwdwedqwedweqdewd}
 and set $p^{\mathrm{lf}} \coloneqq c\circ  p$.}
 \begin{ddd}\label{troigjwrtoijgergrweg}
The transformation $p$ in \eqref{qwoeifjoifjewfqwefwfqewfq} is called the Paschke {transformation}.
 \end{ddd}
 
The proof of Assertion  \ref{qreoigjoergegqrgqerqfewf}.\ref{weiufhiqwefewwfqewfewf} will be finished in Section \ref{eroigheifogegegergewgerg}, and  the proof of Assertions   \ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf}  and   \ref{qreoigjoergegqrgqerqfewf}.\ref{wrtigjiogowrefwerfewrf}   will be completed in Section \ref{wefgiojwoegergregegw}.
Once $p$ is constructed, which is not at all trivial, the verification that it is an equivalence under additional conditions follows the route described above, i.e.\ by reducing it to the case of orbits.  The verification that $p$ is indeed an equivalence on $G$-orbits with finite stabilizers also turns out to be quite involved and uses a lot of the properties of the $K$-theory functor for $C^{*}$-categories obtained in \cite{cank}. 


In the case of  trivial coefficients and under the assumption of the existence of an absorbing representation $(H,\rho,\phi)$ we can compare the version of the Paschke morphism $p^{(H,\rho,\phi)}_{X}$ from \eqref{rfqwpofkjpqowefewfeqwfqef} with the newly defined Paschke morphism $p_{(X,\tau)}$ from \eqref{Paschke-morphisms} (in particular their domains): Indeed, in Proposition \ref{iuhugiergwergergwg} we show that there is a commutative diagram
\[ \begin{tikzcd}
	K_{\bC}^{G,\cX}(X) \ar[r,"\gamma"] \ar[d,"{p_{(X,\tau)}}"] & \mathrm{KK}(\C,Q^{G}(X))\ar[d,"{p_{X}^{(H,\rho,\phi)}}"]  \\
	K_{\bC}^{G,\mathrm{An}} (X) & \ar[l,"\simeq"]  \mathrm{KK}^{G}(C_{0}(X) ,Q(H))  
\end{tikzcd}\]
so that, under the assumption that $p_{(X,\tau)}$ is an equivalence, $\gamma$ is an equivalence if and only if $p_X^{(H,\rho,\phi)}$ is.


 


\subsection*{Assembly maps}


Our original motivation to show the Paschke duality theorem above
was the wish to write out a complete proof for the fact the  homotopy theoretic assembly map  of Davis--L\"uck \cite{davis_lueck}
and the analytic assembly map appearing in the Baum--Connes conjecture 
  are equivalent. 
 Such an equivalence was asserted in \cite{hamped}, but the details of the proof given in this reference
 remained sparse. {While we were preparing this paper, a comparison of the two assembly maps was recently also carried out by Kranz \cite{kranz} with methods different from ours, see Remark~\ref{Remark:comparison-assembly}.}
 
 Homotopy theoretic assembly maps are generally defined for any equivariant homology theory $G\mathbf{Orb}\to \mathbf{M}$ with 
 cocomplete target $\mathbf{M}$ and a family $\cF$ of subgroups, see  Definition~\ref{woiejtgoiwegegrgergwer}. Our comparison concerns
 the functor  
 \begin{equation}\label{asdvijqoirgfvfsdvsva}
K\bC^{G} \colon G\mathbf{Orb}\to \Sp^{\mathrm{}} \, , \quad S\mapsto  K\bC\cX^{G}_{c,G_{can,min}}(S_{min,max})\, ,
\end{equation}
see Definition \ref{qroeigjoqergerqfewewfqewfeqwf}. 
 Note that the twist is different from the one used in the Definition~\eqref{qewfoijfoiqwefqewfewfqewfd} of $K^{G,\cX}_{\bC}$, namely {it is  $G_{can,min}$ rather than  $G_{can,max}$}.
 {For appropriate choice of coefficients $(\bC,{\mathbf{K}})$, the  functor $K\bC^{G}$ is  equivalent to the functor introduced by Davis--L\"uck, 
 Remark~\ref{oiejgoigsvfdvfvs}.}


The equivariant homology theory $K\bC^{G}$ gives rise to a canonical functor
$$K\bC^{G} \colon G\Top\to \Sp^{\mathrm{}}$$ denoted by the same symbol, see Definition \ref{weiogwegerffs}. 
For any family of subgroups $\cF$ of $G$  the homotopy theoretic  assembly map {can be described as the map}
\[\mathrm{Asmbl}^{h}_{E_{\cF}G^\mathrm{CW}} \colon K\bC^{G}(E_{\cF}G^\mathrm{CW})\to K\bC^{G}(*)\] 
induced by the projection $E_{\cF}G^\mathrm{CW}\to *$, where $E_{\cF}G^\mathrm{CW}$ is a $G$-CW-complex representing the homotopy type of the classifying space of $G$ with respect to the family $\cF$. 


{If $\cF\subseteq \mathbf{Fin}$, then} we define  
\[ RK^{G,\mathrm{An}}_{\bC} (E_{\cF}G^\mathrm{CW}) \coloneqq \colim_{W\subseteq E_{\cF}G^\mathrm{CW} } K^{G,\mathrm{An}}_{\bC}(W),\]
where the colimit runs over the $G$-finite subcomplexes of {$E_{\cF}G^\mathrm{CW}$. We }construct an 
 analytic assembly map \begin{equation}\label{eqwfqwefewfwdqwedqwq}
 \mathrm{Asmbl}^{\mathrm{an}}_{\cF} \colon RK^{G,\mathrm{An}}_{\bC} (E_{\cF}G^\mathrm{CW}) \to  \Sigma \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)\, ,
\end{equation}
 where the $C^{*}$-category  $ {\mathbf{K}}^{(G)}_{\mathrm{std}}$ is defined in Definition \ref{qeroighjoreqgreqfefwfqwef}  and the reduced crossed product for $C^{*}$-categories is as introduced in \cite{cank}, see Section~\ref{weroigjwoergergwgre} for the precise construction. 


In the 
  special case where  $A$ is a  {unital} separable $G$-$C^*$-algebra and the coefficient pair is $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}_{c}^{G}({A}))$ the assembly map  $\mathrm{Asmbl}^{\mathrm{an}}_{\cF}$ 
 reduces to the classical one. 
More precisely,  
 in Definition \ref{wergoijowergerrrfrfrfrfeggwgw} we provide a spectrum-valued version of  the classical Kasparov assembly map \begin{equation}\label{rerwwerfefefwef}
\mu^{Kasp}_{\cF,A}:RK^{G,\mathrm{an}}_{A}(E_{\cF}G^\mathrm{CW})\to \mathrm{KK}(\C,A\rtimes_{r}G)\ ,
\end{equation}
{and in Proposition \ref{weiojgwoerferww} we show the following comparison result.}
 \begin{prop} We have an equivalence between the assembly maps 
$ \mathrm{Asmbl}^{\mathrm{an}}_{\cF} $ from \eqref{eqwfqwefewfwdqwedqwq}
and $\mu^{Kasp}_{\cF,A}$  from \eqref{rerwwerfefefwef}.
\end{prop}


{We consider the spectrum-valued refinement {\eqref{rerwwerfefefwef}} of Kasparov's assembly map  as an interesting result in its own right. {In view of the definition of the domain, one has to construct  a family of such  assembly maps  indexed by the $G$-finite subcomplexes $W$
 of $E_{\cF}G^\mathrm{CW}$ which is {compatible}   with inclusions}. While it is easy to lift Kasparov's construction to a map of spectra for each such $W$ individually, and it is {also} easy to obtain the {required} compatibility on the level of homotopy groups, it is a non-trivial matter {to enhance the   compatibility to}  the spectrum level. {We obtain this enhancement in the form of the natural transformation \eqref{rgfewrfwweferf}.}
 } 
 
The following theorem (whose proof will be completed in Section \ref{wtoiguwegwegr3gergreggrgergregwe}) now provides a comparison of the Davis--L\"uck and  Baum--Connes assembly maps on the level of homotopy groups.  As indicated earlier, a version of this result has recently been shown also by \cite{kranz} with completely different methods.

 \begin{theorem}\label{wtoiguwegwergergregwe}
If we assume that  ${\mathbf{K}}$ is compatible with orthogonal sums
(Definition \ref{wefiqrjhfoifewqdqwedqwedqewd}), then 
we have  a commuting square
\begin{equation}\label{avvoijfoivjoafvasddsvdvasdv}
\xymatrix{
K\bC_{*}^{G}(E_{\cF}G^\mathrm{CW})\ar[rr]^-{\pi_{*}\mathrm{Asmbl}_{\cF}^{h}} && K\bC_{*}^{G}(*)\ar[d]^{\cong}\\
RK^{G,\mathrm{An}}_{\bC,*+1} (E_{\cF}G^\mathrm{CW})\ar@{-}[u]^{\cong}\ar[rr]^{\pi_{*+1}\mathrm{Asmbl}^{\mathrm{an}}_{\cF}} && \mathrm{KK}_{*}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)
}
\end{equation}
   \end{theorem}      The left vertical equivalence  in \eqref{avvoijfoivjoafvasddsvdvasdv} is, in a non-obvious manner, a consequence of our Paschke duality Theorem~\ref{qreoigjoergegqrgqerqfewf}. 
{If $A$ is a unital  $G$-$C^{*}$-algebra, then the ideal ${\mathbf{K}}:=\mathbf{Hilb}_{c}^{G}(A)$ in $\bC:=\mathbf{Hilb}^{G}(A)$ is compatible with orthogonal sums by  Proposition \ref{qroigoergergwggrgwerg} and hence satisfies the assumption of the Theorem \ref{wtoiguwegwergergregwe}.}
   
   
{We believe that our method can be upgraded to provide a commutative diagram on the spectrum level, but carrying this out would involve to control further large coherence diagrams. We refrain from doing this additional step at this point, but emphasize that the passage to a statement about homotopy groups is really only in the very final step where one filters $E_{\cF}G^\mathrm{CW}$ through $G$-finite subcomplexes. For any $G$-finite $X$ in place of $E_{\cF}G^\mathrm{CW}$, the diagram in Theorem~\ref{wtoiguwegwergergregwe} commutes already before applying homotopy groups. {In particular,  the square in \eqref{avvoijfoivjoafvasddsvdvasdv}}
commutes before applying homotopy groups when there is a $G$-finite model of $E_\cF G^\mathrm{CW}$. It is just that we have not worked out that the homotopies for varying $X$ can be obtained in a compatible way. This problem is not visible to homotopy groups, and hence one obtains Theorem~\ref{wtoiguwegwergergregwe} irrespective of this issue.}
    
    
We note that it is important to consider the reduced crossed product in the target for the approach presented here. 
While the construction of the analytic assembly map easily lifts to the maximal crossed product our method unfortunately does not generalize to produce the corresponding comparison of assembly maps also for the maximal crossed product. 
 
 
 
   
Finally, we explain some relations to previous works on (equivariant) Paschke duality and the analytical assembly map.
We begin with  Paschke duality.

  
  \begin{rem} 
 {Valette established} a non-commutative generalization of the classical Paschke duality \cite{Val83}  {whose statement we briefly recall here}. 


 We  consider  a $C^{*}$-algebra $B$ with a strictly positive element. Then we have an exact sequence
 $$0\to B\otimes K(\ell^{2})\to \cM^{s}(B)\stackrel{\pi}{\to}  \mathcal{Q}^{s}(B)\to 0\ ,$$ where $ \cM^{s}(B)$ is the stable multiplier algebra and the stable Calkin algebra   $\mathcal{Q}^{s}(B)$ is defined as the quotient.
  In place of $\phi\colon C_{0}(X)\to B(H)$ above we now consider a
 unital separable nuclear $C^{*}$-algebra $A$ with a representation
  $\tau\colon A \to  B(\ell^{2})$ such that $\tau(A)\cap K(\ell^{2})=\{0\}$ and set
  $\phi\colon A\stackrel{1\otimes \tau}{\to}  \cM^{s}(B)\stackrel{\pi}{\to} \mathcal{Q}^{s}(B)$.
  We further  replace $Q(H,\phi)$ from above by  the commutant 
  $Q(A,\phi,B)\coloneqq\phi(A)'$ of the image of $\phi$.  The proof of the following result employs Kasparov's generalization of Voiculescu's Theorem.
  \begin{prop}[{\cite[Prop. 3]{Val83}}]
  We have an isomorphism
  $$KK_{*}(\C,Q(A,\phi,B))\cong KK_{*-1}(A,B)$$
  which is natural in $A$ and $B$.
  \end{prop}
 In this statement  $KK_{*}$ denote Kapsarov's $KK$-groups. Note  that  
  the right-hand side in the  original statement of Valette is   expressed  in terms of $Ext$-groups
  which are isomorphic to the $KK_{*}$-groups under the given assumptions on $A$ and $B$.
 If $B$ is in addition  $\sigma$-unital, then by \cite[Prop.\ 1.19]{KKG} the $KK$-group on the right-hand side   coincides with the 
 $KK$-group  obtained from the spectrum-valued $KK$-theory constructed in \cite{KKG}.
 \end{rem}
  
  



 
\begin{rem}\label{Remark:previous-work}
Our Theorem \ref{qreoigjoergegqrgqerqfewf}  is similar in spirit to  \cite[Thm.\ 1.5]{Benameur:2020aa}.   
But while Theorem \ref{qreoigjoergegqrgqerqfewf}   produces a natural transformation between spectrum-valued functors which becomes an equivalence when evaluated on spaces satisfying suitable finiteness conditions,
 \cite[Thm.\ 1.5]{Benameur:2020aa} states an isomorphism between $K$-theory groups for a fixed space. While the class of spaces to which 
  \cite[Thm.\ 1.5]{Benameur:2020aa} applies is larger than  the class of spaces for which 
   Theorem \ref{qreoigjoergegqrgqerqfewf} provides an equivalence, our theorem allows to treat more general coefficients.

   
But even in the case where 
both theorems are applicable the   technical details of their statements are quite different so that at the moment it is difficult to compare them   in a precise way.  
In the following we explain this problem in greater detail.

The space $X$ in \cite[Thm.\ 1.5]{Benameur:2020aa}\footnote{denoted by $Z$ in the reference} is a  metric space with an isometric proper  cocompact action of $G$.
In order {to} fit into our theorem we must require in addition that 
 it admits a scale $\tau$ such that $(X,\tau)$ belongs to $\mathbf{UBC}^{\mathrm{scl}}$,
  and that it is homotopy equivalent to a $G$-finite $G$-simplicial complex.
  The domain of the Paschke map  in    \cite[Thm.\ 1.5]{Benameur:2020aa} is the $K$-theory of a certain $C^{*}$-algebra $Q^{G}(H,\rho,\phi)$, where $H$ is a sufficiently large  Hilbert $C^{*}$-module over a commutative unital $C^{*}$-algebra $A$.  In order to compare with our theorem
  we would restrict the coefficients to the special case $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}^{G}_{c}(A))$.
  We then could ask whether {we have}
  $$K^{C^{*}\mathrm{Cat}}_{*}(\mathbf{Q}^{G}_{\bC}(X))\cong K^{C^{*}\mathrm{Alg}}_{*}(Q(H,\rho,\phi))\, ,$$
  see \eqref{gepojgregergegwgwergrweg}. 
  The construction of a comparison map could proceed similarly as the construction
  of the map $\gamma$ in Proposition \ref{iuhugiergwergergwg} once we know that 
  $(H,\rho,\phi)$ is absorbing in the sense of   the natural generalization of Definition \ref{werogihetgergfdsf} to controlled Hilbert $A$-modules.
 
 On the positive side, in the case $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}^{G}_{c}(A))$
 the targets  of the two Paschke duality   
   maps in    \cite[Thm.\ 1.5]{Benameur:2020aa} and \ref{qreoigjoergegqrgqerqfewf}
   are equivalent in view of 
   $$K^{G,\mathrm{An}}_{\bC}(X)\simeq \mathrm{KK}^{G}(C_{0}(X),\mathbf{Q}^{(G)}_{\mathrm{std}})
 \stackrel{\text{Prop.\ }\ref{wegojeogrregregwergwegre}}{\simeq} \Sigma KK^{G}(C_{0}(X),A)
$$
provided $X$ is homotopy equivalent to a $G$-finite $G$-CW-complex.
   \hspace*{\fill}$\qed$
\end{rem}

\begin{rem}\label{Remark:comparison-assembly}
As mentioned earlier, in \cite{kranz} Kranz also provides an identification of the Davis--L\"uck assembly map and the Kasparov assembly map. In fact, the contribution of Kranz is a comparison of    the Davis--L\"uck assembly map  with the version of the assembly map  introduced by    Meyer--Nest \cite{MR2193334}.  The latter is compared in  \cite{MR2193334} with Kasparov's assembly map employing work of 
Chabert--Echterhoff \cite{MR1836047}.  In Section \ref{thelast} 
 we will give a detailed account of   the argument of Kranz  using the  $\infty$-categorical language of equivariant $\mathrm{KK}$-theory developed in \cite{KKG}. 
 As an application, we will show in  Propositions \ref{eqriogergreqfqwfefq} and   \ref{weiojgwoerferww} that the  Kasparov assembly map   is an equivalence for compactly induced coefficient categories or algebras.
  \hspace*{\fill}$\qed$\end{rem}


\section{Constructions with \texorpdfstring{$\boldsymbol{C^{*}}$}{Cstar}-categories}\label{section:constructions}

In order to fix size issues we choose a sequence of four Grothendieck universes whose sets will be called very small, small, large,  and very large, respectively.  The group $G$, bornological coarse spaces or $G$-topological spaces,   etc.\ belong to  
the very small universe.  The categories of these objects,   the coefficient  $C^{*}$-categories,  the categories of controlled objects, and the 
values {of}  the $K$-theory functor $K^{C^{*}\mathrm{Cat}}$ will
belong to the small universe. The categories of spectra $\Sp$ and $\mathrm{KK}^{G}$
are large, but locally small. They are objects of a category of stable $\infty$-categories $\mathbf{CAT}^{ex}_{\infty}$ which is itself very large.

  



Both versions of  {$K$-homology} considered in the present paper depend on the choice of a small unital coefficient $C^{*}$-category
$\bC$ with $G$-action and an invariant ideal ${\mathbf{K}}$.  We consider the ideal as implicitly associated to $\bC$ and follow the convention from \cite{coarsek} to
  drop it    from the notation.   

\begin{ex}\label{weriogwergw}Our basic example is the category $\bC=\mathbf{Hilb}^{G}(A)$ of Hilbert $C^{*}$-modules over a very small
$C^{*}$-algebra $A$ with $G$-action (see \cite{cank}  for a detailed description) and the ideal ${\mathbf{K}}=\mathbf{Hilb}^{G}_{c}(A)$ of compact operators. It follows from \cite[Sec.\ 17]{cank}   that in this special case
the functor $K\bC^{G}$ in \eqref{asdvijqoirgfvfsdvsva} is equivalent to the functor $K^{G}_{A}$ considered in \cite{kranz} which directly generalizes the construction in \cite{davis_lueck} from $\C$ to general $G$-$C^{*}$-algebras $A$.
\hspace*{\fill}$\qed$
\end{ex}


In this subsection we describe various $C^{*}$-categories derived  from this pair $(\bC,{\mathbf{K}})$.

First we fix the notation convention concerning the $G$-action on $  \bC$.
If $C$ is an object of $  \bC$ and $g$ is in $G$, then we let $g(C)$ denote the object 
obtained by applying $g$ to {$C$}. 
Similarly, if $A$ is a morphism in $  \bC$, then we write
$g(A)$ for the morphism obtained by applying $g$ to~$A$.


\begin{ddd}\label{tgijrogwegweffsfv}
A $G$-object in $  \bC$ is a pair $(C,\rho)$ of an object in $\bC$ and a family $\rho=(\rho_{g})_{g\in G}$ of unitaries $\rho_{g} \colon C\to g(C)$ such that $g(\rho_{h})\rho_{g}=\rho_{gh}$ for all $h,g$ in $G$.
\end{ddd} 

\begin{ex}
If $G$ acts trivially on $\bC$, then the datum of a $G$-object $(C,\rho)$ in $\bC$ is the same as an object  $C$ of $\bC$ with a homomorphism $\rho \colon G\to \Aut_{\bC}(C)$, $g\mapsto \rho_{g}$, such that
$\rho_{g^{-1}}=\rho_{g}^{*}$.
\hspace*{\fill}$\qed$
\end{ex}


\begin{ddd}\label{qegijqgiofjfqewfqwef}
The category of $G$-objects in $  \bC$ is 
the $C^{*}$-category with $G$-action   $  \bC^{(G)}$ in $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})$ defined as follows:
  \begin{enumerate}
  \item objects: The objects of   $  \bC^{(G)}$ are    the $G$-objects in $  \bC$.
  \item morphisms: The morphisms in $  \bC^{(G)}$ are given by 
  \begin{equation}\label{vewvejvovwrwervwevwe}
  \Hom_{\bC^{(G)}}((C,\rho),(C',\rho')) \coloneqq \Hom_{\bC}(C,C')\, . 
\end{equation}
  \item composition and involution: The composition and involution are inherited from $\bC$.
  \item $G$-action:
  \begin{enumerate}
  \item objects: $G$ fixes the objects of $  \bC^{(G)}$.
  \item morphisms: $g$ in $G$ acts on a morphism   $A \colon (C,\rho)\to (C',\rho')$ by
 \begin{equation}\label{eqwfijoifqjowiefjwoefewqfewqf}
g\cdot A \coloneqq \rho_{g}^{\prime, -1} g(A )\rho_{g}\, .
\end{equation}
  \end{enumerate}
 \end{enumerate}
 \end{ddd}
 
 Note that we use the notation $g\cdot-$ in order to denote the $G$-action on morphisms between $G$-objects which should not be confused with the original action denoted by $g(-)$.
 
\begin{rem}
In this remark we provide a categorical interpretation of the construction of the category $  \bC^{(G)}$ of $G$-objects in $  \bC$.
 We have a chain of inclusions
 $${\mathbf{C}^{\ast}\mathbf{Cat}}\to {\mathbf{C}^{\ast}\mathbf{Cat}}^{(G)}\to \Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})$$
 where $ {\mathbf{C}^{\ast}\mathbf{Cat}}^{(G)}$ is the full subcategory of $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})$
 of  $C^{*}$-categories with strict $G$-action   fixing  the objects.
 We further consider ${\mathbf{C}^{\ast}\mathbf{Cat}}$ as a full subcategory of $ {\mathbf{C}^{\ast}\mathbf{Cat}}^{(G)}$ of $C^{*}$-categories on which $G$ acts trivially. 
 We will consider these categories as $2$-categories whose $2$-morphisms are unitary isomorphisms of functors. Then
 these inclusions have right-adjoints, i.e., we have a diagram of $2$-categorical adjunctions
 $$\xymatrix{{\mathbf{C}^{\ast}\mathbf{Cat}}\ar@/_-1cm/[rr]^-{\mathrm{incl}} &\perp&{\mathbf{C}^{\ast}\mathbf{Cat}}^{(G)}\ar@/_-1cm/[rr]^-{\mathrm{incl}}\ar@/_-1cm/[ll]^-{  \bC^{(G)}\mapsto  \bC^{G}}&\perp&\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}}) \ar@/_-1cm/[ll]^-{  \bC\mapsto   \bC^{(G)}}\,.}$$ 
 Here $\bC^{G}$ is the wide subcategory of $  \bC^{(G)}$  of $G$-invariant morphisms.
 \hspace*{\fill}$\qed$
  \end{rem}
   
    
 
 The category $  \bC$ is unital, but in general it  contains  objects whose identity morphism does not belong to the ideal $  {\mathbf{K}}$. \begin{ddd}\label{wefbsfopkfvdfsv} We let $  {\mathbf{K}}^{u}$
denote  the full subcategory of unital objects in $  {\mathbf{K}}$, i.e., objects whose identities belong to ${\mathbf{K}}$.
\end{ddd}
Note that $  {\mathbf{K}}^{u}$ is unital and carries an induced $G$-action.   
  
  \begin{ex}
  Let $A$ {be} a unital $C^{*}$-algebra with $G$-action, and consider $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}_{c}^{G}(A))$ as in Example \ref{weriogwergw}. Then
  ${\mathbf{K}}^{u}=\mathbf{Hilb}^{G}(A)^{\proj, \mathrm{fg}}$ is the full subcategory of $\mathbf{Hilb}^{G}(A)$ of finitely generated projective Hilbert $A$-modules.
  \hspace*{\fill}$\qed$
  \end{ex}

 
  A $G$-bornology  on a $G$-set $X$ 
is a $G$-invariant  subset of $\mathcal{P}_{X}$ which is closed 
under forming finite unions, subsets and contains all one-point subsets. 
 A $G$-bornological space
 is a pair $(X,\cB)$  of a $G$-set  $X$ with   a $G$-bornology $\cB$ whose elements will be called the bounded subsets of $X$. If $ (X,\cB)$ and $ (X',\cB')$ are $G$-bornological spaces
  and  $f \colon X\to X'$ is an equivariant map of underlying $G$-sets, then $f$ is called  proper
 if  $f^{-1}(\cB')\subseteq \cB$.
  By  $G\mathbf{Born}$ denote the category of very small $G$-bornological spaces and proper maps.
We refer to \cite{equicoarse} for more details. We will usually use the shorter  notation $X$ for $G$-bornological spaces. To any $G$-set $S$ we can associate the the following objects in $G\mathbf{Born}$.
\begin{enumerate}
\item $S_{min}$ is $S$ equipped with the minimal bornology consisting of the finite subsets.
The map $S\mapsto S_{min}$ is functorial for morphisms of $G$-sets with finite fibres.
\item $S_{max}$ is $S$ equipped with the maximal bornology consisting of all subsets of $S$. We   have a functor $G\Set\to G\mathbf{Born}$ given on objects by $S\mapsto S_{max}$. 
\end{enumerate}


Let $X$ be in $G\mathbf{Born}$. 
 
 \begin{ddd}\label{oiguoeigqregregwergregwgrg}
 A subset $L$ of $X$ is called locally finite if $B\cap L$ is finite for every bounded subset in $X$.
 \end{ddd}

     
In the following definition we use the notion of orthogonal sums of objects in a $C^{*}$-category as developed in \cite{cank}.\footnote{Note that this notion coincides with the one in \cite{fritz} but differs from the {one} in \cite{antoun_voigt}.} 
  
\phantomsection\label{qroigjqoifjmefmewf89uqf8wefqwefqf}
Let $p$ be an orthogonal projection on an object $C$ in a $C^{*}$-category, i.e., an endomorphism of $C$ satisfying $p^{*}=p$ and $p^{2}=p$.
A morphism $u \colon C'\to C$ represents the image of $p$ if $u$ is an isometry (this means that $u^{*}u=\id_{C'}$) and $uu^{*}=p$.
We say that $p$ is effective if it admits an image.\label{effective_projection} In the present paper we will not consider non-orthogonal projections, and therefore we will omit the word orthogonal from now on.
 
 The following definition is taken from \cite{coarsek}. {Let $X$ be in $G\mathbf{Born}$.}

 \begin{ddd}\label{qeroigergeggweerg} A locally finite $X$-controlled $G$-object in $  \bC$ is a triple
$(C,\rho,  \mu)$, where:
\begin{enumerate}
\item  $(C,\rho)$ is an object in $  \bC^{(G)}$.
\item\label{eoigeogerwgrwerg}  $ \mu$ is an invariant, finitely additive, effective projection-valued measure on $X$ with values in $\End_{\bC}(C)$ such that the following properties hold:
\begin{enumerate}
\item  $\mu(X)=\id_{C}$.
\item\label{ogureoiguogr9g02034} $\mu(\{x\})\in {\mathbf{K}}$ for all $x$ in $X$.
\item\label{qeroighjoergfqewfqf} $C$ is isomorphic to the {orthogonal} sum of the images of the family of projections $(\mu(\{x\}))_{x\in X}$.
\item\label{toiwjjkfmfdfds89w} $\supp(\mu)$ is locally finite.
\end{enumerate}
\end{enumerate}
\end{ddd} 

\begin{rem} \label{rqfqiofioewjeowfewfqewfqwefe} In this remark we provide some details concerning Condition \ref{eoigeogerwgrwerg}. It first of all says that 
 $\mu$ is a function from the power set $\mathcal{P}_{X}$ of $X$ to the set of  projections in $\End_{\bC}(C)$ such that for all   $Y,Z$  in $\mathcal{P}_{X}$ with $Y\subseteq Z$  we have $\mu( Z)=\mu(Y)+\mu(Z\setminus Y) $. The adjective ``effective'' means that
  for every $Y$ in $\mathcal{P}_{X}$ the projection $\mu(Y)$ admits an image. This is automatic if we assume that $\bC$ is idempotent complete.
     The invariance condition of $\mu$ means that
\begin{equation}\label{wfoizeqwfu98u9e8wfqewf}
g\cdot \mu(Y)=\mu(gY)
\end{equation}  for all $g$ in $G$ and subsets $Y$ of $X$.  The support of $\mu$ is the subset
$$\supp(\mu) \coloneqq \{x\in X\:|\: \mu(\{x\})\not=0\}$$ of $X$.  

The Conditions \ref{toiwjjkfmfdfds89w}, \ref{qeroighjoergfqewfqf} and \ref{ogureoiguogr9g02034} together imply that $\mu(B)$ belongs to ${\mathbf{K}}$ for every bounded subset $B$ of $X$.
 
 The Condition \ref{qeroighjoergfqewfqf}  can be understood as follows.
 Note that a morphism $A \colon C'\to C$ gives rise to a family $(A_{x})_{x\in X}$
  of morphisms $ A_{x} \coloneqq \mu(\{x\})A \colon C'\to C$ such that $ \|\sum_{x\in Y} A_{x}^{*}A_{x}\|\le \|A\|^{2}$ for all finite subsets $Y$ of $X$. 
The Condition \ref{qeroighjoergfqewfqf} is equivalent to the converse: Assume  that we are given a family $(B_{x})_{x\in X}$ of morphisms  $B_{x} \colon C'\to C$  with $\mu(\{x\})B_{x}=B_{x}$ for all $x$ in $X$  such that
 $$\sup_{Y\subseteq X, |Y|<\infty}\|\sum_{x\in Y} B_{x}^{*}B_{x}\| <\infty\, .$$
Then Condition \ref{qeroighjoergfqewfqf} requires that
 there exists a unique morphism $B \colon C'\to C$ such that $B_{x}=\mu(\{x\})B$ for all $x\in X$. 
 
 Using the involution we can give  an equivalent characterization using morphisms from $C$.
 \hspace*{\fill}$\qed$
\end{rem}

Recall that $X$ is in $G\mathbf{Born}$.

   \begin{ddd}\label{qrohwoifewfqwefqwfewqf}
   We define the unital $C^{*}$-category  with $G$-action ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)$ 
   as follows:
   \begin{enumerate}
   \item objects: The objects of  ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)$  are the  locally finite $X$-controlled $G$-objects in $  \bC$.
   \item morphisms: The morphisms in  ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)$ are given by $$\Hom_{{\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)}((C,\rho,  \mu),(C',\rho',  \mu')) \coloneqq \Hom_{  \bC^{(G)}}((C,\rho),(C',\rho'))\, .$$
   \item composition, involution and $G$-action: The composition, involution and  the $G$-action are induced from $  \bC^{(G)}$. 
   \end{enumerate}
   \end{ddd}

 
 
 We have a fully faithful forgetful functor
 \begin{equation}\label{qewfqwefefqewfqfef}
\cF \colon {\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)\to   \bC^{(G)}\, , \quad (C,\rho,\mu) \mapsto (C,\rho)\, .
\end{equation}
 
\begin{ddd}\label{qeroighjoreqgreqfefwfqwef}\mbox{}
\begin{enumerate}
\item We define the unital  $C^{*}$-category   with $G$-action  $  \bC_{\mathrm{std}}^{(G)}$  
as the full subcategory of $  \bC^{(G)}$
of objects which  {are} isomorphic to objects of the form $\cF( (C,\rho,\mu))$ for some object
$(C,\rho,\mu)$ in ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(Y_{min})$ for some free $G$-set $Y$.
\item \label{wgiowjegoreregwg}We   let $  {\mathbf{K}}_{\mathrm{std}}^{(G)}$ be the $G$-invariant  ideal   of $  \bC_{\mathrm{std}}^{(G)}$ of morphisms belonging to ${  {\mathbf{K}}}$. 
\item \label{wrtogwprtgwergwergfreferf}We define  the unital  $C^{*}$-category   with $G$-action
\begin{equation}\label{wefwfwfwefflkjlqwepofp}
  \mathbf{Q}^{(G)}_{\mathrm{std}} \coloneqq \frac{  \bC_{\mathrm{std}}^{(G)}}{  {\mathbf{K}}_{\mathrm{std}}^{(G)}}\, .
\end{equation} 
\end{enumerate}
\end{ddd} 
 
\begin{rem} 
The Definition \ref{qeroighjoreqgreqfefwfqwef}.\ref{wgiowjegoreregwg} makes sense in view of \eqref{vewvejvovwrwervwevwe} since  ${  {\mathbf{K}}}$ is a $G$-invariant wide subcategory of ${  \bC}$.
\hspace*{\fill}$\qed$
 \end{rem}
 
 

\section{\texorpdfstring{$\boldsymbol{G}$}{G}-bornological coarse spaces and  \texorpdfstring{$\boldsymbol{K\bC\cX^{G}_{c}}$}{KCXGc}}


 
 
 {As in Section~\ref{section:constructions}}, we fix a pair  $(   \bC,   {\mathbf{K}})$  of a  {unital} $C^{*}$-category with $G$-action $   \bC$ and an invariant ideal $   {\mathbf{K}}$.  
 In this section we recall the construction 
 of the equivariant coarse homology theory 
$$K\bC\cX^{G}_{c} \colon G\mathbf{BC}\to \Sp^{\mathrm{}}$$ introduced in \cite{coarsek} which will give rise to the second version of the analytic $K$-homology  {$K_\bC^{G,\cX}$} below {(Definition \ref{qriofjqofewfefqeqff}).} 
In order to define $K\bC\cX^{G}_{c}$ the pair  $(   \bC,   {\mathbf{K}})$ must satisfy the following two conditions:
\begin{ass}\label{wergoiewgewrergwrgwgwg}
\mbox{}
\begin{enumerate}
\item $\bC$ is countably additive.
\item\label{wegoijwegregwergwegwer} $\bC$ is idempotent complete. 
\end{enumerate}
\end{ass}

Subsets of $X\times X$ will be called entourages on $X$.
The set $\mathcal{P}_{X\times X}$  of all entourages is a monoid with involution, where
the composition of entourages $U$ and $V$ is  the entourage $$U\circ V \coloneqq \pr_{14}[(U\times V)\cap (X\times \diag(X)\times X)]\, ,$$
the unit is the entourage $\diag(X)$, and the involution is given by the formula $$U^{*} \coloneqq \{(y,x)\mid (x,y)\in U\}\, .$$
The monoid $\mathcal{P}_{X\times X} $ acts on $\mathcal{P}_{X}$ by
\begin{equation}\label{wfwfwefsdfsdfd}
(U,Y)\mapsto U[Y] \coloneqq \pr_{1}[U\cap (X\times Y)]\, .
\end{equation} 
A $G$-coarse structure $\cC$ on a $G$-set $X$ is by definition  a $G$-invariant submonoid of $\mathcal{P}_{X\times X}$ which is  
closed under taking subsets and forming finite  unions, and in which the subset of $G$-invariant entourages $\cC^{G}$ is cofinal with respect to the inclusion relation.
A $G$-coarse space is a pair $(X,\cC)$ of a $G$-set and a $G$-coarse structure. If $ (X,\cC)$  and $ (X',\cC')$ are two $G$-coarse spaces and $f \colon X\to X'$ is an equivariant map of the  underlying $G$-sets, then $f$ is controlled if $(f\times f)(\cC)\subseteq \cC'$. Finally, 
a coarse structure $\cC$ is compatible with a bornology $\cB$ if
$\cC[\cB]\subseteq \cB$.

The category $G\mathbf{BC}$ of $G$-bornological coarse spaces was introduced in \cite{equicoarse}. Its objects are triples $(X,\cC,\cB)$ of a very small 
$G$-set $X$ with a $G$-coarse structure $\cC$     and a $G$-bornology $\cB$ which is compatible with $\cC$. Morphisms are maps of $G$-sets which are controlled   and proper.
 We usually use the shorter  notation $X$ for $G$-bornological coarse spaces.
 We refer to \cite{equicoarse} and \cite{buen} for details about this category.



Let $X$ be in $G\mathbf{BC}$. Then we can consider the category \begin{equation}
\mathbf{C}^{G}_{\mathrm{lf}}(X) \coloneqq \lim_{BG} {\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)
\end{equation}
in ${\mathbf{C}^{\ast}\mathbf{Cat}}$ {(cf.\ Definition~\ref{qrohwoifewfqwefqwfewqf} for  ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)$).}
 It is the wide subcategory of $ {\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)$ consisting of the $G$-invariant morphisms, i.e., morphisms $A$ satisfying $g\cdot A=A$ for all $g$ in $G$, see Definition \ref{qegijqgiofjfqewfqwef} and \eqref{eqwfijoifqjowiefjwoefewqfewqf}.
 Note that this construction does not  use the coarse structure yet, but this will be the case in the following. 
  
If $Y,Y'$ are two subsets of $X$ and $U$ is an entourage of $X$, then we say that $Y'$ is $U$-separated from $Y$ if $Y'\cap U[Y]=\emptyset$, see \eqref{wfwfwefsdfsdfd} for the definition of the $U$-thickening $U[Y]$ of $Y$. 
We say that  a morphism $A \colon (C,\rho,\mu)\to (C',\rho',\mu')$ in $ \mathbf{C}^{G}_{\mathrm{lf}}(X)$ is $U$-controlled if
$\mu'(Y')A\mu(Y)=0$ for all pairs of subsets $Y',Y$ of $X$ such that $Y'$ is $U$-separated from $Y$.

\begin{ddd}\label{rfquhwfiuqwhfiufewqefqwefqwefwefqwef}
We define  $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ in ${\mathbf{C}^{\ast}\mathbf{Cat}}$ as follows:
\begin{enumerate}
\item objects: The objects of $  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ are the objects of $\mathbf{C}^{G}_{\mathrm{lf}}(X)$.
\item morphisms: The space of morphisms $\Hom_{  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)}((C,\rho,\mu),(C',\rho',\mu'))$
is the closed subspace of $\Hom_{\mathbf{C}^{G}_{\mathrm{lf}}(X)}((C,\rho,\mu),(C',\rho',\mu'))$ generated by {those} 
morphisms which are $U$-controlled for some coarse entourage $U$ of $X$.
\item\label{weoigjowegergegwrggwergw} composition and involution: The composition and the involution of $  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ {are} inherited from $\mathbf{C}^{G}_{\mathrm{lf}}(X)$. 
\end{enumerate}
\end{ddd}
Note that Point \ref{weoigjowegergegwrggwergw} requires a justification which is given in \cite[Sec.~4]{coarsek}.
 

 \begin{ddd}\label{wtogjoergrwegreggwege}
We define a functor
 $$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}} \colon G\mathbf{BC}\to {\mathbf{C}^{\ast}\mathbf{Cat}}$$  as follows:
 \begin{enumerate}
 \item objects: The functor $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}$ sends $X$ in $G\mathbf{BC}$ to $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ in ${\mathbf{C}^{\ast}\mathbf{Cat}}$.
 \item morphisms: The functor $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}$ sends a morphism $f \colon X\to X'$ {in $G\mathbf{BC}$} to the functor
 $f_{*} \colon  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)\to  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X')$ defined as follows:
 \begin{enumerate}
 \item objects: $f_{*}(C,\rho,\mu) \coloneqq (C,\rho,f_{*}\mu)$.
 \item \label{oiwejgiowegregwr}morphisms: $f_{*}(A) \coloneqq A$.
  \end{enumerate}
\end{enumerate}
\end{ddd}
For the verification that $f_{*}$ is well-defined  
we again refer to  \cite[Sec.~4]{coarsek}. It is at this point where we need the Assumption \ref{wergoiewgewrergwrgwgwg}.\ref{wegoijwegregwergwegwer}. 
Following \cite[Def.~1.8]{KKG} we let $\mathrm{KK}$ denote the presentable stable $\infty$-category 
 representing the finitary extension of 
Kasparov's $K\!K$-theory from separable to all small $C^{*}$-algebras. 
We furthermore consider the  functor $\mathrm{kk}_{C^{*}\mathbf{Cat}} \colon C^{*}\mathbf{Cat}^{\mathrm{nu}} \to \mathrm{KK}$ defined in  \cite[Def.\ 1.27]{KKG}. We then define  
 the topological $K$-theory functor for $C^{*}$-categories by
\begin{equation}\label{rewgwergwergegwergrewgregwregegwre}
K^{C^{*}\mathrm{Cat}} \coloneqq \mathrm{KK}(\C,-) \colon C^{*}\mathbf{Cat}^{\mathrm{nu}}\to \Sp^{\mathrm{}}\, ,
\end{equation} 
where we use the abbreviation $$\mathrm{KK}(-,-) \coloneqq \map_{\mathrm{KK}}(\mathrm{kk}_{C^{*}\mathbf{Cat}}(-),\mathrm{kk}_{C^{*}\mathbf{Cat}}(-)) \colon (C^{*}\mathbf{Cat}^{\mathrm{nu}})^{\mathrm{op}}\times C^{*}\mathbf{Cat}^{\mathrm{nu}}\to \Sp^{\mathrm{}}\, .$$ 
The functor  \eqref{rewgwergwergegwergrewgregwregegwre} is equivalent to the functors  considered in  \cite{joachimcat}, \cite[Sec.\ 8.5]{buen}, \cite[Sec.\ 12]{cank}. 
Note that here we consider $C^{*}$-algebras like $\C$ as $C^{*}$-categories with a single object.

\begin{ddd}\label{qrogijeqoifefewfefewqffe}
We define the functor $K\bC\cX^{G}_{c}$ as the composition
$$K\bC\cX^{G}_{c} \colon G\mathbf{BC}\xrightarrow{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}} {\mathbf{C}^{\ast}\mathbf{Cat}}\xrightarrow{K^{C^{*}\mathrm{Cat}}} \Sp^{\mathrm{}}\, .$$
\end{ddd}

For the definition of the notion of an equivariant coarse homology theory and its additional properties we refer to \cite{equicoarse}. The following  theorem is shown in \cite[Sec.~6]{coarsek} (and  \cite[Sec.~11]{coarsek} for strong additivity).


\begin{theorem}\label{qeroigqwefqewfefewfq}
 $K\bC\cX^{G}_{c}$ is an equivariant coarse homology theory which is in addition strong, continuous and strongly additive.
 \end{theorem}
 
   
\section{\texorpdfstring{$\boldsymbol{G}$}{G}-uniform bornological coarse spaces, cones and \texorpdfstring{$\boldsymbol{K_{\bC}^{G,\cX}}$}{KGXC}}\label{woigjowteggwergewrgerg}


A $G$-uniform structure on $X$ is a $G$-invariant subset ${\mathcal{U}}$ of $\mathcal{P}_{X\times X}$ consisting of entourages  containing the diagonal,  which
 is closed under taking supersets, finite intersections and the involution, and which has the property that every $U$ in ${\mathcal{U}}$ contains a $G$-invariant element of ${\mathcal{U}}$ and  admits $V$ in ${\mathcal{U}}$ with $V\circ V\subseteq U$.  A $G$-uniform space is a pair  $(X,{\mathcal{U}})$ of a $G$-set and a $G$-uniform structure.
 Let $(X,{\mathcal{U}})$ and  $(X',{\mathcal{U}}')$ be $G$-uniform spaces and $f \colon X\to X'$ be a $G$-invariant map of the underlying sets. Then $f$ is uniform  if $(f\times f)^{-1}({\mathcal{U}}')\subseteq {\mathcal{U}}$.
  A uniform structure ${\mathcal{U}}$ is compatible with a coarse structure if ${\mathcal{U}}\cap \cC\not=\emptyset$.

Let $G\mathbf{UBC}$ denote  the category of $G$-uniform bornological coarse spaces introduced in \cite{equicoarse}.  Objects are tuples $(X,\cC,\cB,{\mathcal{U}})$ such that $(X,\cC,\cB)$ is a $G$-bornological coarse space and ${\mathcal{U}}$ is a $G$-uniform structure  compatible with $\cC$. Morphisms are morphisms of $G$-bornological coarse spaces which are in addition uniform.  We will usually use the shorter  notation $X$ for $G$-uniform bornological coarse spaces.
We have  canonical forgetful functors 
\begin{equation}\label{qweflkjoqwfeqewfwefeefeqwf}
G\mathbf{UBC}\to G\mathbf{BC}\ , \quad G\mathbf{UBC}\to G\Top
\end{equation} 
 which forget the uniform structure or take the underlying $G$-topological space, respectively.

If not said differently we will consider all subsets of $\R^{n}$ for some $n$ as objects of $G\mathbf{UBC}$ with the 
  trivial $G$-action and the structures induced by the standard metric.

    
   The categories $G\mathbf{BC}$ and $G\mathbf{UBC}$ have monoidal structures $\otimes$ which  {are} the cartesian structure on the underlying $G$-uniform and $G$-coarse spaces (see   \cite[Ex.\ 2.17]{equicoarse} for the case of $G\mathbf{BC}$) such that the forgetful functor 
   $G\mathbf{UBC}\to G\mathbf{BC}$ is symmetric monoidal in the canonical way. 
   The bornology on {$X \otimes X'$} is generated by the subsets $B\times B'$ for all bounded subsets $B$ of $X$ and $B'$ of $X'$, respectively.  
   
    
    
    
    
Let $X$ be in $G\mathbf{UBC}$.    
\begin{ddd}
$X$ is flasque if  it is a retract of $[0,\infty)\otimes X$. 
\end{ddd}

Note that this definition is a little more restrictive than the definition given in \cite[Text before Def.\ {3.10}]{ass}. 
The same argument as for   \cite[Lem.\ 3.28]{buen}  in {the} non-equivariant case shows {that} 
  the  underlying $G$-bornological coarse space of $X$ is flasque in the generalized sense.
   
The notion of homotopy in the category $G\mathbf{UBC}$ is defined in the usual   manner using the interval 
functor $X\mapsto [0,1]\otimes X$. 
   
Recall the definitions of uniformly or coarsely excisive pairs from   \cite[Def.\ 3.3]{ass}  and  \cite[Def.\ 3.5]{ass}.

   
Let $E \colon G\mathbf{UBC} \to \mathbf{M}$ be a functor whose target is a stable $\infty$-category.
\begin{ddd}\label{qeroigjoerwgwergwrgwe9111}
\mbox{}
\begin{enumerate}
\item $E$ is homotopy invariant if it sends the projection $[0,1]\otimes X\to X$ to an equivalence for every $X$ in $G\mathbf{UBC}$.
\item $E$ satisfies closed excision if  $E(\emptyset)\simeq 0$ and for every uniformly   and coarsely excisive pair $(Y,Z)$ of invariant closed subsets of some $X$ in $G\mathbf{UBC}$
such that $X=Y\cup Z$ the square
$$\xymatrix{E(Y\cap Z)\ar[r]\ar[d]&E(Y)\ar[d]\\mathbb{E}(Z)\ar[r]&E(X)}$$
is a push-out square.
\item $E$ vanishes on flasques if $E(X)\simeq 0$ for any flasque $X$ in $G\mathbf{UBC}$.
\item $E$ is $u$-continuous if for every $X$ in $G\mathbf{UBC}$ we have $ \colim_{V} E(X_{V})\simeq  E(X) $, where $V$ runs over $\cC^{G}\cap {\mathcal{U}}$,  
and $X_{V}$ is obtained from $X$  by  replacing its coarse structure $\cC$   on $X$   by the coarse structure   generated by $V$.
\end{enumerate}
\end{ddd}

 
 In the following we describe two variants of  the  cone functor from $G\mathbf{UBC}$ to $G\mathbf{BC}$.


Let $X$ be in $G\mathbf{UBC}$ with uniform structure ${\mathcal{U}}$.  
Note that ${\mathcal{U}}$ and ${\mathcal{P}_{X\times X}}$ are posets with respect to the inclusion relation.

\begin{ddd} \label{wfiuhiufqwefeewqfqwfewfqewf}
A    scale for $X$ is a pair  $\tau \coloneqq (\phi, \psi)$,  where: \begin{enumerate} \item 
  $\phi \colon \R\to (0,\infty)$ is a bounded, non-increasing function such that $\phi_{|(-\infty,0]}\equiv 1$.
   \item  
$\psi \colon \R\to \mathcal{P}(X\times X)^{G}$ is a non-increasing function with the following properties:
\begin{enumerate}
\item If $t$ is in $(-\infty,0]$, then $\psi(t)=X\times X$.
\item \label{gpowgergerlgewrgwregerg} For every   $V$  in ${\mathcal{U}}$ there exists $t_{0}$ in $\R$ such that $\psi(t)
 \subseteq V$ for all $t$ in $[t_{0},\infty)$.
\end{enumerate}
 \end{enumerate} 
\end{ddd}

\begin{ddd}\label{qriugiqregergwerg}
We define the geometric cone-at-$\infty$ of $X$   to be the object ${\mathcal{O}}^{\infty}(X)$ in  $G\mathbf{BC}$ given as follows:
\begin{enumerate}
\item The underlying $G$-set of ${\mathcal{O}}^{\infty}(X)$ is $\R\times X$.
\item The bornology of ${\mathcal{O}}^{\infty}(X)$ is generated by the subsets $[-r,r]\times  B$ for all $r$ in $(0,\infty)$ and bounded subsets $B$ of $X$.
\item\label{qoirgjoqierfewfqfewfw} The coarse structure is  generated by the  entourages $U\cap U_{\tau}$ for all scales $\tau \coloneqq (\phi, \psi)$ with $\lim_{t\to \infty} \phi(t)=0$,   where   $U$ is a coarse entourage of $\R\otimes X$ and
\begin{align}
\mathclap{
U_{\tau} \coloneqq \{((s,x),(t,y))\in (\R\times X)\times ( \R\times X)\:|\: |s-t|\le \phi(\max\{s,t\}), (x,y)\in \psi(\max\{s,t\})\}\,.
}\notag\\
\label{fqewfpokopqkfqef}
\end{align}
 \end{enumerate}
We furthermore define the cone ${\mathcal{O}}(X)$  of $X$ to be the subset $[0,\infty)\times X$ with the induced structures.
\end{ddd}

In particular, in the case of $X=*$ the choice of $\psi$ is irrelevant and we consider
\begin{equation}\label{vweiojeoivjweervervwver}
U_{\phi} \coloneqq \{(s,t)\in \R\:|\: |s-t|\le \phi(\max\{s,t\})\}\, .
\end{equation}
 


\begin{ddd}\label{wepogjpweggreewg}
We define functors $${\mathcal{O}}^{\infty},{\mathcal{O}} \colon G\mathbf{UBC}\to G\mathbf{BC}$$ as follows:
\begin{enumerate}
\item objects: The functors send $X$ in $G\mathbf{UBC}$ to ${\mathcal{O}}^{\infty}(X)$ or ${\mathcal{O}}(X)$, respectively.
\item \label{qwoifhoirfqfefqwef} morphisms: The functors send a morphism $f \colon X\to X'$ in $G\mathbf{UBC}$ to the morphism
${\mathcal{O}}^{\infty}(X)\to {\mathcal{O}}^{\infty}(X')$ or ${\mathcal{O}} (X)\to {\mathcal{O}} (X')$ given by
$\id_{\R}\times f$ or $\id_{[0,\infty)}\times f$, respectively.
\end{enumerate}
\end{ddd}

The definition of the functors for morphisms in Point \ref{qwoifhoirfqfefqwef} needs a justification which is given e.g.\ by  \cite[Lem.\ 5.15]{buen}.

 

    For $X$ in $G\mathbf{UBC}$ we have a natural sequence of maps in $G\mathbf{UBC}$ \begin{equation}\label{ecelkmlwqcqwecwecqcqwecq}
X\to {\mathcal{O}}(X)\to {\mathcal{O}}^{\infty}(X)\to \R\otimes X
\end{equation}
 called the cone sequence. Here the first map is given by $x\mapsto (0,x)$, the second map is the inclusion, and the third map is the identity on the underlying sets.


 Let $E \colon G\mathbf{BC}\to \mathbf{M}$ be a functor with target a stable $\infty$-category.
Then we consider the functors
\begin{align}\label{wqefqwefewfqewf}
E{\mathcal{O}}^{\infty} &\coloneqq E\circ {\mathcal{O}}^{\infty} \colon G\mathbf{UBC}\to \mathbf{M} \\
 E{\mathcal{O}}& \coloneqq E\circ {\mathcal{O}} \colon G\mathbf{UBC}\to \mathbf{M} \, . \nonumber
\end{align}
\begin{prop}\label{qroigjqowfewfqwefqw}
We assume that $E$ is a coarse homology {theory} which is in addition strong.
Then the functors $E{\mathcal{O}}^{\infty}$ and $E{\mathcal{O}}$ have the following properties:
\begin{enumerate}
\item homotopy invariance,
\item closed excision,
\item vanishing on flasques,
\item $u$-continuous.
\end{enumerate}
Moreover, the cone sequence  \eqref{ecelkmlwqcqwecwecqcqwecq} induces a  fibre sequence of functors
\begin{equation}\label{wefqwefqewewfwdqewdewdqede}
E\to E{\mathcal{O}}\to E{\mathcal{O}}^{\infty}\xrightarrow{\partial^{{\mathtt{Cone}}}} \Sigma E\, .
\end{equation}
\end{prop}
This proposition follows from the results stated in \cite[Sec.~9]{ass} (which are stated there in the non-equivariant case, but the same proof applies here). In particular, the list of properties of the functors is given by  \cite[Lem.~9.6]{ass} and 
the cone sequence follows from  \cite[(9.1)]{ass}. 
Note that we consider $E$ {in \eqref{wefqwefqewewfwdqewdewdqede}} as a functor on $G\mathbf{UBC}$ by using the forgetful functor \eqref{qweflkjoqwfeqewfwefeefeqwf}.  

  Let $Y$ be in $G\mathbf{BC}$ and $E \colon G\mathbf{BC}\to \mathbf{M}$  be some functor.  \begin{ddd}[{{\cite[(10.17)]{equicoarse}}}]\label{regoiergowregrwergwreg} We define the twist $E_{Y}$ of $E$ by $Y$ as the functor
 $$E_{Y} \colon G\mathbf{BC}\to \mathbf{M}\, , \quad E_{Y}(X) \coloneqq E(X\otimes Y)\, .$$
 \end{ddd} 
 
 The following has been shown in \cite[{Lem.\ 4.17 \& 11.25}]{equicoarse}:
 \begin{lem}
 If $E$ is a coarse homology theory, then so is its twist $E_{Y}$.
 If $E$ is strong, then so is $E_{Y}$.
 \end{lem}
 
 We apply this construction to the equivariant coarse homology theory $K\bC\cX^{G}_{c}$ from  Definition \ref{qrogijeqoifefewfefewqffe}. 
  The group $G$ gives rise to the $G$-bornological coarse spaces
$G_{can,min}$ and $G_{can,max}$  \cite[Ex.\ 2.4]{equicoarse}.
Here $min$ and $max$ refer to the minimal (finite subsets) and maximal (all subsets) bornologies, and the canonical coarse structure {$can$}
is the minimal $G$-coarse structure such that $G_{can}$ is a connected $G$-coarse space. It is generated by the entourages
$\{(g,h)\}$ for all {$(g,h)$ in $G \times G$.} 
Later we will in particular consider the coarse homology theories
$K\bC\cX^{G}_{c,G_{can,max}}$ and $K\bC\cX^{G}_{c,G_{can,min}}$ obtained from 
$K\bC\cX^{G}_{c}$ by twisting (Definition \ref{regoiergowregrwergwreg}) with $G_{can,max}$ and $G_{can,min}$.



 
\begin{ddd}\label{qriofjqofewfefqeqff} 
We define the functor
$$K_{\bC}^{G,\cX} \colon G\mathbf{UBC}\to \Sp^{\mathrm{}}$$
as the composition
$$ K_{\bC}^{G,\cX} \colon G\mathbf{UBC}\xrightarrow{{\mathcal{O}}^{\infty}} G\mathbf{BC} \xrightarrow{K\bC\cX^{G}_{c,G_{can,max}}} \Sp^{\mathrm{}}\,.$$
\end{ddd}


The following proposition   lists the properties of the functor $K_{\bC}^{G,\cX}$.
 It is a consequence of Theorem \ref{qeroigqwefqewfefewfq} and Proposition \ref{qroigjqowfewfqwefqw}. It should be compared with 
 the properties of the functor $K_{\bC}^{G,\mathrm{An}}$  from \eqref{vfdsvsvvsvsdvdsadsvdsva} which are    listed in \cite[Prop.\ 1.10]{KKG}.
 \begin{prop}\label{oirgjoerwgwegregw9}
The functor $K_{\bC}^{G,\cX}$ has the following properties:
\begin{enumerate}
\item closed excision,
\item homotopy invariant,
\item $u$-continuous,
\item vanishing on flasques.
\end{enumerate}
\end{prop}
  
 
\section{Preparation of the statement of Theorem  \ref{qreoigjoergegqrgqerqfewf}}
 
In this section we will introduce the constructions going into the statement  {of} our main Theorem  \ref{qreoigjoergegqrgqerqfewf}
about Paschke duality  relating the functors
$K_{\bC}^{G,\mathrm{An}}$ and $K_{\bC}^{G,\cX}$ {with each other.} Note that these functors are defined on different domains, namely $G\mathbf{LCH}^{\mathrm{prop}}_{+}$  {(recall that this is the category  of all locally compact $G$-spaces and partially defined proper maps)} and $G\mathbf{UBC}$. Hence before we can    construct a natural transformation  between these functors we  must restrict both functors to a common domain.   

Let $X$ be in $G\mathbf{UBC}$.

\begin{ddd}
We define $\tilde {\mathcal{O}}^{\infty}(X)$ and $\tilde {\mathcal{O}}(X)$ in $G\mathbf{BC}$ as in Definition \ref{qriugiqregergwerg} but dropping the condition that $\lim_{t\to\infty} \phi(t)=0$ in   Definition \ref{qriugiqregergwerg}.\ref{qoirgjoqierfewfqfewfw}.
\end{ddd}

Let $X$ be in $G\mathbf{UBC}$ and let $\tau=(\phi,\psi)$ be a scale for $X$ (Definition \ref{wfiuhiufqwefeewqfqwfewfqewf}).
\begin{ddd}
We say that $\tau$ is uniform   if $\psi$ takes values in the uniform structure of $X$.
\end{ddd}

Note that if the uniform structure of $X$   does not admit a countable basis, then $X$ does not admit any uniform scale. 

Let $\tau$ be a uniform scale on $X$ and recall the  definition \eqref{fqewfpokopqkfqef} of $U_{\tau}$. 
\begin{ddd}\label{qegoiejrgoeirgqwefqwfwqefqewf}\mbox{}
\begin{enumerate}
 \item
We let ${\mathcal{O}}^{\infty}_{\tau}(X)$ be the $G$-bornological coarse space
obtained from ${\mathcal{O}}^{\infty}(X)$ by replacing its coarse structure by the   coarse structure generated by all entourages of the form $U\cap U_{\tau}$ for all coarse entourages $U$ of $\R\otimes X$. 
\item\label{weigoggsfgsfgsfgsdfgsg} We let ${\mathcal{O}}_{\tau}(X)$ be the subset $[0,\infty)\times X$ with the structures induced from ${\mathcal{O}}^{\infty}_{\tau}(X)$.
\end{enumerate}
\end{ddd}

 
Let $\tau,\tau'$ be uniform scales for $X$ and assume that $\tau\ge \tau'$, i.e., $\phi\ge \phi'$ and $\psi\supseteq \psi'$. Then 
the identity of the underlying sets   are  morphisms
\begin{align}\label{fqewfwfqewfwfewfewf}
&{\mathcal{O}}^{\infty}_{\tau'}(X) \to{\mathcal{O}}_{\tau}^{\infty}(X)\to\tilde  {\mathcal{O}}^{\infty}(X)\leftarrow {\mathcal{O}}^{\infty}(X)  \\& {\mathcal{O}}_{\tau'}(X) \to {\mathcal{O}}_{\tau} (X)\to \tilde {\mathcal{O}} (X)\leftarrow {\mathcal{O}}(X) \nonumber
\end{align}
 in $G\mathbf{BC}$.
  
 \begin{prop}\label{eqroguerogeregwergwre}
 The morphisms in \eqref{fqewfwfqewfwfewfewf} are sent to equivalences by every coarse homology theory.
 \end{prop}
\begin{proof}
The proofs of the results in \cite[Sec.~8]{ass} apply also to the equivariant case here without change. 
\end{proof}
 
 
 Let $X$ be in $G\mathbf{UBC}$ and
  $\tau$ be a uniform  scale for $X$. 
  Then we can form ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ in $G\mathbf{BC}$.
  Let $L$ be a subset of ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$.
\begin{ddd}\label{ewriogwergwregwref}
$L$ is locally uniformly   finite  if for every
bounded subset $B$ of $X$ and coarse entourage $U$ of $  {\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ {we have}
$$ \sup_{y\in  [0,\infty)\times B \times G }  |L\cap U[\{y\}]|  <\infty\, .$$  
\end{ddd}


\begin{ddd}\label{qreoighwoefewqffqewffewfq}
We define the category $G\mathbf{UBC}^{\mathrm{scl}}$ as follows:
\begin{enumerate}
\item objects: The objects of $G\mathbf{UBC}^{\mathrm{scl}}$
are pairs $(X,\tau)$ of  $X$ in $G\mathbf{UBC}$  and  a   {uniform}  scale $\tau$ such that:
\begin{enumerate}
\item The underlying topological space of $X$
is locally compact Hausdordd and second countable.
\item  The bornology  of $X$ consists of  the relatively compact subsets.
\item \label{fhewquifhqwieufhiqwefqwef}${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ admits an invariant locally uniformly  finite subset $L$ such that $L\to {\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ is a coarse equivalence.
\end{enumerate}
 \item \label{rogijtoigjwegerwergwerwregwerg}A morphism $f \colon (X,\tau)\to (X',\tau')$ is a morphism $f \colon X\to X'$ in $G\mathbf{UBC}$ such that  there exists an integer $m$ such that $(f\times f)\circ \psi\subseteq (\psi')^{m}$ and $U_{\phi}\subseteq U_{\phi'}^{m} $ (see \eqref{vweiojeoivjweervervwver} for notation), where $\tau=(\phi,\psi)$ and $\tau'=(\phi', \psi')$.
\end{enumerate}
\end{ddd}  

Here for  an entourage $U$ we let $\smash{U^{m} \coloneqq \underbrace{U\circ\dots \circ U}_{m\text{-times}}}$ denote the $m$-fold composition of $U$ with itself. 

\begin{rem}\label{rem_scale_needed}
The reason for working with fixed scales and not with $\tilde {\mathcal{O}}(X)$ or ${\mathcal{O}}(X)$ is that we do not expect that
$\tilde {\mathcal{O}}(X)\otimes G_{can,max}$ or ${\mathcal{O}}(X)\otimes G_{can,max}$  satisfy the analogue of  Condition   \ref{qreoighwoefewqffqewffewfq}.\ref{fhewquifhqwieufhiqwefqwef}.
But this condition  is crucial for the construction of the Paschke morphism. It goes into the proof of   Lemma \ref{wtioghjwergfgrefwref} below. 
\hspace*{\fill}$\qed$
\end{rem}

In the following discussion we provide some  examples for objects in $G\mathbf{UBC}^{\mathrm{scl}}$. 
 The existence of such objects is not completely obvious because of    Condition   \ref{qreoighwoefewqffqewffewfq}.\ref{fhewquifhqwieufhiqwefqwef}.

Let $G\mathbf{Simpl}$ be the category of $G$-simplicial complexes {and equivariant simplicial maps.} 
 We will consider $G$-simplicial complexes as metric spaces with the  spherical path quasi-metric. The latter is uniquely characterized by the property that it induces the spherical metric on simplices, i.e., the metric induced from the Riemannian metric on the standard spheres $S^{n}$ 
by identifying  the $n$-simplices with the intersection  $S^{n}\cap [0,\infty)^{n+1}$.
  
 
 
 \begin{ddd}
 A $G$-simplicial complex has bounded geometry if there is a uniform bound on  the number of simplices in the stars of the {vertices.} 
 \end{ddd}
 
{Note that if a $G$-simplicial complex $K$ has bounded geometry, then $K$ is locally finite and finite-dimensional.}
 
 We let $G\mathbf{Simpl}_{{\mathrm{bg}}}^{\mathrm{prop}}$ be the subcategory of  $G\mathbf{Simpl}$  of $G$-simplicial complexes of bounded geometry and proper maps.
 
  
 Let $X$ be a metric space with an isometric $G$-action. For $r$ in $[0,\infty)$ and $x$ in $X$ we let $B_{r}(x)$ denote the {closed} metric ball of radius $r$ with center $x$.
 \begin{ddd}\label{qroighqffeqfewfqfewf}
We say $X$ has locally uniformly  bounded geometry if it admits a family of  invariant subsets $(Y_{s})_{s\in (0,\infty)}$ such that:    \begin{enumerate}
 \item  \label{rghdrgbgdgd}
 $Y_{s}$ is $s$-dense in $X$, i.e., $\bigcup_{y\in Y} B_{s}(y)=X$. 
 \item\label{qregqgfwefqweeqwfq} For every $n$ in $\nat$ and compact subset $B$ of $X$ we have \begin{equation}\label{asdccsdcadscdscdc}
 \sup_{s\in (0,\infty)} \sup_{x\in X}  |Y_{s}\cap B\cap B_{ns}(x)|<\infty\,. 
\end{equation}
  \end{enumerate}
 \end{ddd}
 This definition is weaker  than
 \cite[Def.\ 4.1]{Engel:2019aa} since we  
 require  the uniformity conditions \eqref{asdccsdcadscdscdc} only on compact subsets of $X$.
 
\begin{ex}\label{wtogijtwroggregwgerwgec}
A complete Riemannian manifold with isometric $G$-action 
has locally uniformly bounded geometry.  The argument of \cite[Lem.\ 5.2]{Wulff:2016aa} 
applies without the bounded geometry assumption {on the manifold} since we restrict to compact subsets.

The objects of $G\mathbf{Simpl}^{\mathrm{prop}}_{{\mathrm{bg}}}$ have  locally uniformly bounded geometry.
Given $s$ in $(0,\infty)$,   the  subset $Y_{s}$  can be chosen as
zero skeleton of a sufficiently iterated barycentric subdivision such that Condition \ref{qroighqffeqfewfqfewf}.\ref{rghdrgbgdgd}  is satisfied. Condition \ref{qroighqffeqfewfqfewf}.\ref{qregqgfwefqweeqwfq} is then automatic by the bounded geometry assumption.
 \hspace*{\fill}$\qed$
\end{ex}



 If $(X,d)$ is a quasi-metric space, then we consider the function
 $$\psi_{d} \colon [0,\infty)\to \mathcal{P}(X\times X)$$ given by 
 $$\psi_{d}(t) \coloneqq \{(x,x')\in X\times X\:|\:  d(x,x')\le t^{-1}\}\, .$$ 
 Then  $\tau_{d} \coloneqq ({\mathtt{const}}_{1},\psi_{d})$ is a uniform scale.
 We will equip metric spaces with the uniform bornological coarse structure induced by the metric.
 \begin{prop}\label{qwfiuhqwifweewfefffqedqwedew} \mbox{}
   \begin{enumerate} 
 \item \label{qrgiohjiowfeweqfefefqdqwdwe} If
$(X,d)$ is a  proper metric space with an isometric $G$-action and  locally uniformly 
  bounded geometry,   
 then $(X,\tau_{d})$ belongs to $G\mathbf{UBC}^{\mathrm{scl}}$.
 \item \label{ergoijeogwegregwgergwgwerg}There is a functor 
 $G\mathbf{Simpl}^{\mathrm{prop}}_{{\mathrm{bg}}} \to  G\mathbf{UBC}^{\mathrm{scl}}$ which sends
 $X$ in $ G\mathbf{Simpl}^{\mathrm{prop}}_{{\mathrm{bg}}}$ to $(X,\tau_{d})$. \end{enumerate}
\end{prop}
  \begin{proof} In order to show Assertion \ref{qrgiohjiowfeweqfefefqdqwdwe} 
  the only non-obvious condition to check is that 
  $(X,\tau_{d})$  satisfies \ref{qreoighwoefewqffqewffewfq}.\ref{fhewquifhqwieufhiqwefqwef}.
 We define the $G$-invariant subset $L$ of $[0,\infty)\times X\times G$ as follows. Let $(Y_{s})_{s\in (0,\infty)}$ be a family of subsets of $X$ as in Definition \ref{qroighqffeqfewfqfewf}.
For $n$ in $\nat\setminus\{0\}$ we choose the $ n^{-1}$-dense subset $L_{n,0} \coloneqq Y_{n^{-1}} $ of $X$.
Then we define the $G$-invariant subset  $$L_{n} \coloneqq \{(g\ell,gh)\:|\: g,h\in G \text{ and } \ell\in L_{n,0}\}$$
of $X\times G$. Finally we define the $G$-invariant subset
  $$L \coloneqq \bigcup_{n\in \nat\setminus\{0\}}\{n\}\times L_{n}$$
  of $[0,\infty)\times X\times G$.
  We consider the entourage 
  \begin{align*}
  V_{m} \coloneqq \{((s,x),(s',x'))\in   {([0,\infty)\times X)^2} \:|\: |s-s'|\le m, d(x,x')\le m \min\{ 1, s^{-1},s^{\prime,-1} \}\}
  \end{align*}
  of ${\mathcal{O}}_{\tau}(X)$, and for every finite subset $F$ of $G$ we define the entourage $W_{F} \coloneqq G(F\times F)$ of $G_{can,max}$.  
   
By an inspection we see that
$L$ is $V_{1}\times W_{\{e\}}$-dense.
Since $G$ acts freely on $L$ and {on} ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ it follows that  
the inclusion $L\to {\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ is a coarse equivalence.
  

 It suffices to verify the local uniform finiteness condition {in} Definition  \ref{ewriogwergwregwref} for $L$ {only} for the cofinal set of entourages $V_{m}\times W_{F}$ of ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$, where $F$ runs
 over all finite subsets $F$ of $G$ and $m$ is in $\nat$.
 
 Let $B$ be a compact subset of $X$.
 Let $(s,x,g)$ be in $[0,\infty)\times X\times G$.
 
  Assume that $n$ is the largest integer  smaller than $s-m$.
 If $(s',x',g')$ is in the subset $(V_{m}\times W_{F})[\{(s,x,g)\}]$, then
 $|s'-s|\le m$, $d(x',x)\le m\min\{1, n^{-1}\}$ and $g^{-1}g'\in F^{-1}F$.
 We conclude that $$|(V_{m}\times W_{F})[\{(s,x,g)\}]\cap L\cap ([0,\infty)\times B\times G)|$$ is bounded by
 $|F^{-1}F| \cdot \sum_{k=n}^{n+2m} |Y_{k^{-1}}\cap B\cap  B_{m \min\{1,n^{-1}\}} (x')|$.
 Using Definition \ref{qroighqffeqfewfqfewf}.\ref{qregqgfwefqweeqwfq} we see that  this number is bounded independently of $(s,x,g)$.
 

We now show the  Assertion \ref{ergoijeogwegregwgergwgwerg}.
If $X$ is in $G\mathbf{Simpl}_{{\mathrm{bg}}}^{\mathrm{prop}}$, then $X$ is a metric space with an isometric $G$-action and  locally uniformly bounded geometry.  Therefore 
$(X,\tau_{d})$ is in $G\mathbf{UBC}^{\mathrm{scl}}$.
If $f \colon X\to X'$ is a morphism in $G\mathbf{Simpl}_{{\mathrm{bg}}}^{\mathrm{prop}}$, then 
$f$  is a contraction and proper. Consequently it is a morphism in $G\mathbf{UBC}$. Furthermore,
the conditions in Definition \ref{qreoighwoefewqffqewffewfq}.\ref{rogijtoigjwegerwergwerwregwerg} are satisfied with $m=1$.
\end{proof}





The following lemma  is again not completely obvious because of Condition  \ref{qreoighwoefewqffqewffewfq}.\ref{fhewquifhqwieufhiqwefqwef}.
\begin{lem}
\label{qeroighioerwfqweewfqffew} 
The category $G\mathbf{UBC}^{\mathrm{scl}}$ is closed under taking subspaces with  {the} induced structures and scales. 
\end{lem} 
 \begin{proof}  Let $(X,\tau)$ be in $G\mathbf{UBC}^{\mathrm{scl}}$ and let $Y$ be an invariant subset of~$X$.
 We   consider $Y$ as an object of $G\mathbf{UBC}$ by equipping it with the induced structures.
 Further, we can restrict the uniform scale $\tau$ to a  uniform scale $\tau_{Y}$ on $Y$: if $\tau=(\phi,\psi)$, then $\tau_{Y}=(\phi,\psi\cap(Y\times Y))$.
 We must show that   $(Y,\tau)$ is in $G\mathbf{UBC}^{\mathrm{scl}}$.
 
The only non-trivial part is to check the  condition of Definition \ref{qreoighwoefewqffqewffewfq}.\ref{fhewquifhqwieufhiqwefqwef}, i.e.\  the existence of a locally uniformly finite subset $L_{Y}$ in 
${\mathcal{O}}_{\tau_{Y}}(Y)\otimes G_{can,max}$ such that $L_{Y}\to {\mathcal{O}}_{\tau_{Y}}(Y)\otimes G_{can,max} $ is a coarse equivalence.

By assumption there exists such a subset $L$ in ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$.
Let $U$ be an invariant entourage of ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ such that $L$ is $U$-dense.
For every $G$-orbit  $G\ell$ in $L/G$   represented by $\ell$ in $L$ 
such that $U[\ell]\cap ({\mathcal{O}}_{\tau_{Y}}(Y)\otimes G_{can,max})\not=\emptyset$ we can choose {an} $\ell'$ in $U[\ell] \cap ({\mathcal{O}}_{\tau_{Y}}(Y)\otimes G_{can,max})$.  We then let $L_{Y}$ be the union of these $G$-orbits $G\ell'$. 

Then $L_{Y}$ is $U^{2} $-dense in ${\mathcal{O}}_{\tau_{Y}}(Y)\otimes G_{can,max}$ and {hence} 
$L_{Y}\to {\mathcal{O}}_{\tau_{Y}}(Y)\otimes G_{can,max} $ is a coarse equivalence.


For every bounded subset $B$ of $Y$ and $y$ in $Y$ we have 
$$|L_{Y}\cap B\cap V[\{y\}]|\le |L\cap U[B] \cap UV[\{y\}]|\, ,$$
and this number is uniformly bounded with respect to $y$ in $Y$.
\end{proof}



We have  canonical forgetful functors
\begin{equation}\label{q3oigfherqoifqrfvffqwefqeeqwfqeeqwfqefqefeeqfeqfewffefqewqewfqfqewf}
\iota^{\mathrm{top}} \colon G\mathbf{UBC}^{\mathrm{scl}}\to G\mathbf{LCH}^{\mathrm{prop}}_{+} \quad \text{ and } \quad \iota^{\mathrm{scl}} \colon G\mathbf{UBC}^{\mathrm{scl}}\to G\mathbf{UBC}
\end{equation}
which send $(X,\tau)$ to its underlying $G$-topological space {or} forget the scale, respectively.

Let $(X,\tau), (X',\tau')$ be in $G\mathbf{UBC}^{\mathrm{scl}}$  and $f_{0},f_{1} \colon (X,\tau)\to (X',\tau')$ be two morphisms.  {The following notion of homotopy will be used in Proposition \ref{eoirgjwergerregwgreg} further below.}
\begin{ddd}\label{ergoijogiwregergergwergwegr}
 $f_{0}$ and $f_{1}$ are homotopic if $\iota^{\mathrm{scl}}(f_{0})$ and $\iota^{\mathrm{scl}}(f_{1})$ are homotopic.
\end{ddd}




Recall that Assumption \ref{wergoiewgewrergwrgwgwg} is our standing assumption in order to define $K_{\bC}^{G,\cX}$.
For the Paschke duality theorem we need an additional assumption 
 on  $( \bC,{\mathbf{K}})$ requiring that ${\mathbf{K}}$ is compatible with  orthogonal sums
 which we will explain in the following.
   
   
 
   
   We consider $\bD$ in ${\mathbf{C}^{\ast}\mathbf{Cat}}$ {containing an   ideal $\mathbf{L}$}. 
Let $D$ be an object in $\bD$ and $(e_{i})_{i\in I}$ be a mutually orthogonal family of isometries
$D_{i}\to D$, where $D_{i}$ is belongs to the unital objects $\mathbf{L}^{u}$  of $\mathbf{L}$, see Definition \ref{wefbsfopkfvdfsv} for  $\mathbf{L}^{u}$.  
In \cite[Def.\ 3.7]{cank}, for any $D'$ in $\bD$  we defined the Banach subspace
$\mathbb{K}(D',(D,(e_{i})_{i\in I}))$ of $\Hom_{\bD}(D',D)$ generated by the morphisms of the form
$D'\stackrel{f_{i}}{\to}D_{i}\stackrel{e_{i}}{\to} D$ for some $i$ in $I$.
We now note that 
 $e_{i}f_{i}$  belongs to $\mathbf{L}$. Since $\mathbf{L}$ is closed  
 we have $$\mathbb{K}(D',(D,(e_{i})_{i\in I}))
\subseteq \Hom_{\mathbf{L}}(D',D)\, .$$

\begin{ddd}\label{wefiqrjhfoifewqdqwedqwedqewd}
We say that $\mathbf{L}$ is compatible with orthogonal sums if for every
family $(D_{i})_{i\in I}$ in $\mathbf{L}^{u}$ and pair $(D,(e_{i})_{i\in I})$ representing the orthogonal sum of the family\footnote{in the sense of \cite[Def.\ 3.16]{cank}}
we have
$$\mathbb{K}(D',(D,(e_{i})_{i\in I}))=\Hom_{\mathbf{L}}(D',D)$$
for all $D'$ in $\bD$.
\end{ddd}


Let $    A$ be in $\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$. The following proposition states that the pair
 $$ (\mathbf{Hilb}^{G}(A),\mathbf{Hilb}^{G}_{c}(A))$$ from Example \ref{weriogwergw}
 satisfies all assumptions going into Theorem  \ref{qreoigjoergegqrgqerqfewf}.

\begin{prop}\label{qroigoergergwggrgwerg} \mbox{}
 \begin{enumerate}
\item  $ \mathbf{Hilb}^{G}(A)$ is idempotent complete.
\item $ \mathbf{Hilb}^{G}(A)$ admits all very small sums. 
\item The ideal $\mathbf{Hilb}_{c}^{G}(A)$ is compatible with orthogonal sums.
\end{enumerate}
\end{prop}
\begin{proof}  The category  $ \mathbf{Hilb}^{G}(A)$ admits clearly   very small orthogonal sums
 and is  idempotent complete. We must check that the ideal ${\mathbf{K}} \coloneqq \mathbf{Hilb}_{c}^{G}(A)$ is compatible with orthogonal sums.
 


Let $(C,(e_{i})_{i\in I})$ with $e_{i} \colon C_{i}\to C$  represent the orthogonal sum of the family $(C_{i})_{i\in I}$ in  $\mathbf{Hilb}^{G}(A)$  such that   $C_{i}$ is  in ${\mathbf{K}}^{u}$ 
for all $i$ in $I$. Let $C'$ be any object in $ \mathbf{Hilb}^{G}(A)$. It suffices to show that
$$\Hom_{{\mathbf{K}} }(C',C)\subseteq \mathbb{K}(C',(C,(e_{i})_{i\in I}))\, .$$ 


{Let $\mathrm{Res}^{G}_{1}$ denote the functor which forgets the $G$-action.}
We have a canonical functor  $\iota \colon \mathrm{Res}^{G}_{1}( \mathbf{Hilb}^{G}({A} )) \to \mathbf{Hilb}( {A})$
which sends the objects of $  \mathbf{Hilb}^{G}({A} )$ to the underlying Hilbert $A$-module, and which is the canonical inclusion on morphisms.
By definition the morphisms in ${\mathbf{K}} $ are  precisely  the morphisms in $ \mathbf{Hilb}^{G}({A} )$ which are sent to compact operators by $\iota$.

 

 
 We show for all $c$ in $\iota  C$ and $c'$ in $\iota  C'$ that the one-dimensional operator
$\Theta_{c,c'} \colon \iota C'\to \iota  C$ given by $\xi'\mapsto c\langle c',\xi'\rangle$ belongs to $\mathbb{K}(\iota C',(\iota C,(\iota e_{i})_{i\in I}))$.
Fix $\epsilon$ in $(0,\infty)$. Then we can find a finite subset $I'$ of $I$ such that $$ \tilde c \coloneqq \sum_{i\in I'} e_{i}(e_{i}^{*}(c))$$ satisfies
$$\|c-\tilde c \|\le \frac{ \epsilon}{\|c'\|+1}
\, . $$
 Then $\Theta_{\tilde c,c'}\in  \mathbb{K}(\iota C',(\iota C,(\iota e_{i})_{i\in I}))$ and
$\|\Theta_{c,c'}-\Theta_{\tilde c,c'}\|\le \epsilon$.
 Since  $\mathbb{K}(\iota C',(\iota C,(\iota e_{i})_{i\in I}))$ is closed we conclude that it contains all one-dimensional operators $\Theta_{c,c'}$.
 
 
 
 Let $B$ be in $\Hom_{{\mathbf{K}}}(C',C)$. Then $\iota B$  is compact in the sense of Hilbert $A$-modules.    Given $\epsilon$ in $(0,\infty)$, then since $B$ can be approximated by one-dimensional operators and the latter belong to $\mathbb{K}(C',(C,(e_{i})_{i\in I}))$,
  there is a finite subset $I'$ of $I$ such that $\| B-\sum_{i\in I'}  e_{i} e_{i}^{*} B\|\le \epsilon$.
 
 
  Since $\epsilon$ is arbitrary and $\mathbb{K}(C',(C,(e_{i})_{i\in I}))$  is closed we finally conclude that $B$ belongs to 
  $\mathbb{K}(C',(C,(e_{i})_{i\in I}))$.
 \end{proof}

 


\section{Locality and pseudolocality}\label{qriugfhiergfedfqewfqewf9}


We have a functor $$G\mathbf{LCH}^{\mathrm{prop}} \to G\mathbf{Born}$$ {sending} a  locally compact Hausdorff
$G$-topological space to the underlying $G$-bornological space with the bornology of the relatively compact subsets. 
This functor will usually be dropped from the notation. 
Note that this functor does not extend to the larger category $G\mathbf{LCH}^{\mathrm{prop}}_{+}$  containing maps which are only partially defined.



 For $X$  in $G\Top^{\mathrm{prop}}$ we form  the $G$-$C^{*}$-category 
${\mathbf{C}}^{(G)}_{\mathrm{lf}}([0,\infty)\otimes X\otimes G_{max})$ introduced  in Definition \ref{qrohwoifewfqwefqwfewqf} which {only} depends on the underlying $G$-bornological spaces of $X$. 
 We consider an object $(C,\rho,\mu)$  in 
 ${\mathbf{C}}^{(G)}_{\mathrm{lf}}([0,\infty)\otimes X\otimes G_{max})$.
We  then extend the projection-valued measure $\mu$ to a homomorphism of $C^{*}$-algebras $\phi \colon C_{0}(X)\to \End_{\bC}(C)$
  which sends $f$ in $C_{0}( X)$ to
\begin{equation}\label{ewqfpojlqkrmeflkwefqef}
\phi(f) \coloneqq \int_{ [0,\infty)\times X\times G} \pi^{*}(f) d\mu\, ,
\end{equation}
where
\begin{equation}\label{eq_projection_cone_to_X}
\pi \colon [0,\infty)\times X\times G\to X
\end{equation}
is the projection.
\begin{rem}
This integral   can be interpreted as follows. 
For every $y$ in $[0,\infty)\times X\times G$ we can choose a representative
$u_{y} \colon C_{y}\to C$ of the image of the projection $\mu(\{y\})$ on $C$. By Definition  \ref{qeroigergeggweerg}.\ref{qeroighjoergfqewfqf}
\begin{equation}\label{feqwfpokjfpowefkofqwpeofkoqkkopk}
(C,(u_{y})_{y\in [0,\infty)\times X\times G})
\end{equation}
represents the orthogonal sum of the family $(C_{y})_{y\in [0,\infty)\times X\times G}$. 
Using that $f$ is bounded and that the family $(u_{y})_{y\in [0,\infty)\times X\times G}$ is mutually orthogonal we conclude using  \cite[Cor.~4.1]{cank} that the sum
$$\phi(f):=\sum_{y\in [0,\infty)\times X\times G} u_{y}f(\pi(y)) u_{y}^{*}$$ exists. 
\hspace*{\fill}$\qed$
\end{rem}
 One checks  that $\phi$ is a homomorphism of $C^{*}$-algebras. Furthermore,  using the equivariance \eqref{wfoizeqwfu98u9e8wfqewf} of $\mu$,  one checks that $\phi$ is equivariant in the sense that 
\begin{equation}\label{gwwgegdfsfvfdvsdvsfv}
g^{-1}\cdot \phi(f)=\phi(g^{*}f)
\end{equation}
for all $g$ in $G$, see \eqref{eqwfijoifqjowiefjwoefewqfewqf} for notation.



 
 For the rest of this subsection let $ (X,\tau)$ {be} an object of   the category $G\mathbf{UBC}^{\mathrm{scl}}$ described in Definition \ref{qreoighwoefewqffqewffewfq}.  Recall that 
  the underlying $G$-topological space of $X$ is Hausdorff, locally compact and second countable. 
 We then  have  the cone ${\mathcal{O}}_{\tau}(X)$ in $G\mathbf{BC}$ introduced in Definition \ref{qegoiejrgoeirgqwefqwfwqefqewf}.
Recall  {also} Definition \ref{rfquhwfiuqwhfiufewqefqwefqwefwefqwef} of the $C^{*}$-category 
$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}_{\tau}(X)\otimes  G_{can,max})$.
 
Let $A \colon (C,\rho,\mu)\to (C',\rho',\mu')$ be a morphism in  $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}_{\tau}(X)\otimes  G_{can,max})$.   Our next result states that $A$ is pseudolocal {(in the sense of \cite[Def.~12.3.1]{higson_roe})} when we consider the objects of $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}_{\tau}(X)\otimes  G_{can,max}) $ as $X$-controlled {via \eqref{ewqfpojlqkrmeflkwefqef}.}  
Let   $f$ be in $C_{0}(X)$.

\begin{lem}\label{wtioghjwergfgrefwref}
The difference $A\phi(f)-\phi'(f)A$ belongs to   ${\mathbf{K}}$.
 \end{lem}
\begin{proof}
 

 Assume that $A$ is controlled by a symmetric entourage $U$ of $  {\mathcal{O}}_{\tau}(X)\otimes {G_{can,max}}$, and that 
 $f$ is in $C_{c}(X)$.  We fix $\epsilon$ in $(0,\infty)$.
 
 By Assumption \ref{qreoighwoefewqffqewffewfq}.\ref{fhewquifhqwieufhiqwefqwef} on $X$ there exists a locally uniformly finite subset $L$ (Definition \ref{ewriogwergwregwref}) such that the inclusion
 $i \colon L\to {\mathcal{O}}_{\tau}(X)\otimes  G_{can,max}$ is a coarse equivalence. Therefore  we can choose an inverse morphism
 $j \colon {\mathcal{O}}_{\tau}(X)\otimes  G_{can,max}\to L$ in $G\mathbf{BC}$ such that {$i \circ j$} is $U'$-close to the identity
 of  ${\mathcal{O}}_{\tau}(X)\otimes  G_{can,max}$
 for some coarse entourage $U'$ of ${\mathcal{O}}_{\tau}(X)\otimes  G_{can,max}$.
 This gives an equivariant partition $(Y_{\ell})_{\ell\in L}$ of $ {\mathcal{O}}_{\tau}(X)\otimes  G_{can,max}$, were we set $Y_{\ell} \coloneqq j^{-1}(\{\ell\})$. 
  
    In view of the construction   of the coarse structure on ${\mathcal{O}}_{\tau}(X)$ {in Definition \ref{qriugiqregergwerg}}.\ref{qoirgjoqierfewfqfewfw}     the projection $\pi$ in \eqref{eq_projection_cone_to_X} is controlled, {and hence $(\pi\times \pi)(U'UU')$ a coarse entourage of $X$.} 
Because $\supp(f)$ is compact, it is a bounded subset of $X$. By the compatibility of the bornology and the coarse structure of $X$ its thickening
\begin{equation}\label{eq_defn_thickening_supp_f}
B \coloneqq (\pi\times \pi)(U'UU')[\supp(f)]
\end{equation}
is a bounded subset of $X$, too.
We define the subset $F \coloneqq \pi^{-1}(B)$ of $ {\mathcal{O}}_{\tau}(X)\otimes  G_{can,max}$.
 
 Since $L$  is locally uniformly finite 
we can define the (finite) integer
\begin{equation}\label{wefqewfeewqfqwfwffqfe}
R \coloneqq  \sup_{y\in F}  |L\cap  U'U[\{y\}]  |\, .
\end{equation}
Since $\supp(f)$ is compact and $f$ is continuous, it is uniformly continuous. Hence there exists a uniform entourage 
 $T$ on $X$ such that $$|f(x)-f(x')|\le \frac{\epsilon}{3(1+\|A\|  R)}$$ for all $(x,x')$ in $T$. 

In view of Condition \ref{wfiuhiufqwefeewqfqwfewfqewf}.\ref{gpowgergerlgewrgwregerg} and Definition \ref{qriugiqregergwerg}, the fact that $U^{\prime,2}UU^{\prime,2}$ belongs to the coarse structure of ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ implies that there exists 
$r$ in $(0,\infty)$ sufficiently large such that 
\begin{equation}\label{eq_uniform_cont_f_entourage}
(\pi\times \pi)(U^{\prime,2}UU^{\prime,2} \cap (([0,\infty)\times X\times G)\times ([r,\infty)\times F\times G))\subseteq 
T\, .
\end{equation}
Let $t \colon  [0,\infty)\times X\times G\to [0,\infty)$ be the projection to the first coordinate. 
Assume that   $\ell,\ell'$  are in  $L$ such that  $(\ell',\ell)\in U'UU'$. 
 If  $y'$ is in $Y_{\ell'}$ and $y$ is in $Y_{\ell}$, then $(y',y)\in U^{\prime,2}UU^{\prime,2}$.
 If in addition 
  $\pi(y)\in F$ and $t(y)\ge r$,
then  $(\pi(y'),\pi(y))\in T$ {by \eqref{eq_uniform_cont_f_entourage},}  and hence 
 \begin{equation}\label{vfwpojowpefvfdsvf}
|f(\pi(y))-f(\pi(y'))|\le \frac{\epsilon}{3(1+\|A\|  R)}\, .
\end{equation}
 




We consider the partition of $[0,\infty)\times X\times G$ into the subsets 
$$W \coloneqq \{{y \in [0,\infty)\times X\times G \:|\: r \le t(y)}\} \text{ and }  V \coloneqq \{{y \in [0,\infty)\times X\times G \:|\: 0 \le t(y) < r}\}\, .$$
  We have the following equality:
 \begin{eqnarray}
 A\phi(f)-\phi'(f) A&\stackrel{!}{=}&
A\phi(f)\mu(F)- \phi'(f) A\mu(F)\notag\\
&\stackrel{!!}{=}&
A\phi(f)\mu(F\cap W)+A\phi(f)\mu(F\cap V)\label{eq_A_pseudoloc_terms}\\
&& -  \phi'(f) A\mu(F\cap W) -  \phi'(f) A \mu(F\cap V)\notag
\end{eqnarray}
where for $!$ we use that $\phi(f)\mu(F)=\phi(f)$ since $\supp(f)\subseteq B$ and therefore $\supp(\pi^{*}f)\subseteq F$, and that $A$ is $U$-controlled and 
$\pi^{-1}(X\setminus \supp(f))$ is $U$-separated from $F$ {by \eqref{eq_defn_thickening_supp_f}.}  For $!!$ we use the additivity of $\mu$.
 Since $G_{can,max}$ is bounded  the subset $F\cap V$  is bounded in ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$.
Since $(C,\rho,\mu)$ is locally finite (in particular by the two conditions  \ref{qeroigergeggweerg}.\ref{ogureoiguogr9g02034} and \ref{toiwjjkfmfdfds89w}) the projection $\mu(F\cap V)$ and hence the
morphisms
$A\phi(f)\mu(F\cap V)$ and $  \phi'(f) A \mu(F\cap V)$ belong to ${\mathbf{K}}$.

We now show that the remaining part {of \eqref{eq_A_pseudoloc_terms}} satisfies 
\begin{equation}\label{wecweqwcqsafdcadscdscd}
\|A\phi(f)\mu(F\cap W)- \phi'(f) A\mu(F\cap W)\|\le \epsilon \, .
\end{equation} 
 
 
 For every $\ell$ in $L$ let $v_{\ell} \colon C(\ell)\to C$ be the choice of an image of the projection $\mu(Y_{\ell})$. Since we assume that $\tilde \bC$ is idempotent complete 
 and 
 $(C,\rho,\mu)$ locally finite, by \cite[Lem.\ {4.13}]{cank} the pair  $(C,(v_{\ell})_{\ell\in L})$ represents the orthogonal sum of the family $(C(\ell))_{\ell\in L}$.  We further let
  $(C',(v'_{\ell})_{\ell\in L})$  and $(C'(\ell))_{\ell\in L}$ be a similar choice for $(C',\rho',\mu')$. 
 

 

The morphism $A\phi(f)\mu(F\cap W)- \phi'(f) A \mu(F\cap W)$ is given by the matrix 
$(M_{\ell'\ell})_{\ell',\ell\in L}$ with
$$ M_{\ell'\ell}= v'_{\ell'} [A\phi(f)\mu(F\cap W)- \phi'(f) A \mu(F\cap W)]v^{*}_{\ell}$$
of morphisms from $C$ to $C'$.
For fixed $\ell$ we estimate
the norm of the map
\begin{equation}\label{qwefeqwfqfqewfeqfwefqwefqef}
\sum_{\ell'\in L} M_{\ell'\ell} \colon C\to C'
\end{equation}
by the sum of the norms of the summands. 
If the sum is not zero  we can choose a point $y $ in  $F\cap W\cap Y_{\ell}$. 
If $\ell'$ contributes to this sum, then $(\ell',\ell)\in U'UU'$ and 
$\ell'\in U'U[\{y\}]$.
The estimate \eqref{vfwpojowpefvfdsvf} shows that
$$\|M_{\ell',\ell}\|\le  \frac{\epsilon\|A\|}{(1+\|A\|  R)}\, .$$
 The number of non-zero summands is bounded by $R$ defined in \eqref{wefqewfeewqfqwfwffqfe}.\footnote{The need to have such a bound on the number of summands is the reason for considering scales and  locally uniformly finite subsets.}  
 

 The whole sum in \eqref{qwefeqwfqfqewfeqfwefqwefqef} is  therefore bounded by $\epsilon$.  Using the mutual orthogonality of the family  $ (v_{\ell})_{\ell\in L} $ we conclude 
 the estimate \eqref{wecweqwcqsafdcadscdscd}.
 
Since we can choose $\epsilon$ arbitrarily small and ${\mathbf{K}}$ is closed, we see that
$A\phi(f)-\phi'(f) A\in {\mathbf{K}}$.  


Since ${\mathbf{K}}$ is closed we see that $A\phi(f)-\phi'(f) A \in {\mathbf{K}}$ also for all 
$f$ in $C_{0}(X)$ and $A$ in 
\[\Hom_{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}_{\tau}(X)\otimes G_{can,max})}((C,\rho,\mu),(C',\rho',\mu'))\,.\qedhere\]
\end{proof}

 
In general, if $Z$ is an invariant subset  of a $G$-bornological coarse space $Y$, then we  define the wide subcategory
$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}} (Z\subseteq Y)$ of $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Y)$ (see \cite[Def.\ 5.5]{coarsek}) such that 
 for objects $(C,\rho,\mu)$ and $(C',\rho',\mu')$ in  
$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}
(Z\subseteq Y)$  
$$\Hom_{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}
(Z\subseteq Y)}((C,\rho,\mu),(C',\rho',\mu')) \coloneqq \mu'(Z) \Hom_{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}
(  Y)}((C,\rho,\mu),(C',\rho',\mu'))\mu(Z)\, .$$
Similarly, for an invariant big family ${\mathcal{Z}}=(Z_{i})_{I\in I}$ on $Y$ (see \cite{equicoarse}) we have  the wide subcategory  \begin{equation}\label{f23r8z89fz2893zf892334234f2}
 \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}
({\mathcal{Z}}\subseteq Y) \coloneqq \overline{\bigcup_{i\in I} \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}
(Z_{i}\subseteq Y)}
\end{equation}
  of $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Y)$ (the union and closure are both taken on the level of morphisms). By \cite[Lem.~5.9]{coarsek}  we know that $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}
({\mathcal{Z}}\subseteq Y)$
 is an ideal  in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}
(  Y)$.
 
In our special case  we consider the subsets   
\begin{equation}\label{fqewqewfkqwpefowqefewfef}
Z_{n} \coloneqq [0,n]\times X\times G
\end{equation}
and the big family ${\mathcal{Z}} \coloneqq (Z_{n})_{n\in \nat}$ in ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$.

 
Let $A \colon (C,\rho,\mu)\to (C',\rho',\mu')$ be a morphism in  $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq  {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})$.  Our next result shows that it locally belongs to ${\mathbf{K}}$.   
Let $f$ be in $C_{0}(X)$.
  
\begin{lem}\label{ewiogjoerwgfdsfdgsfg}
We have $\phi'(f)A\in {\mathbf{K}}$ and $A\phi(f)\in {\mathbf{K}}$.
\end{lem}
\begin{proof}
It suffices to show that  $\phi'(f)A\in {\mathbf{K}}$. 
In order to deduce $A\phi(f)\in {\mathbf{K}}$ we {then} use  the involution.
 
 We first assume that  there exists $n$ in $\nat$  such that
$\mu(Z_{n})A \mu(Z_{n})=A$.
If $f$ is in $C_{c}(X)$, then $ \mu(Z_{n} \cap \pi^{-1}(\supp(f)))$ belongs to ${\mathbf{K}}$ since $(C,\rho,\mu)$ is locally finite and $ Z_{n} \cap \pi^{-1}(\supp(f))$ is bounded in ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$. Consequently, {we conclude that}
\[
\phi'(f) A=\phi'(f)\mu'(Z_{n} \cap \pi^{-1}(\supp(f)))A
\]
belongs to ${\mathbf{K}}$.  
Since ${\mathbf{K}}$ is closed we see that $\phi'(f)A\in {\mathbf{K}}$ also for all 
$f$ in $C_{0}(X)$ and $A$ in
\[\Hom_{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})}((C,\rho,\mu),(C',\rho',\mu'))\,.\qedhere\]
\end{proof}

\begin{rem}
Note that the statement of Lemma \ref{ewiogjoerwgfdsfdgsfg} also holds more {generally} for any $X$ in $G\mathbf{UBC}$ and 
${\mathcal{O}}_{\tau}$ replaced by ${\mathcal{O}}$.
\end{rem}

\section{Construction of the Paschke transformation}
 

The main result of the present section is the description of the Paschke morphism for a given space $(X,\tau)$ in $G\mathbf{UBC}^{\mathrm{scl}}$.  
The general idea for its construction via a multiplication map like $\mu_{(X,\tau)}$ as below, {but} with completely different technical details otherwise, has been used at various places, see e.g.\ \cite[Sec.~6.5]{willett_yu_book} or \cite[Sec.~6.4]{wulff_axioms}.
In the Section \ref{wtgiojioergerfwerfewfwerf} we will provide a refinement of this construction to a natural transformation of functors defined on $G\mathbf{UBC}^{\mathrm{scl}}$.

 We start with a description of the following intermediate constructions which go into the  construction of the Paschke morphism:
 \begin{enumerate}
 \item The functor $(X,\tau)\mapsto \mathbf{Q}_{\tau}(X)$ from $G\mathbf{UBC}^{\mathrm{scl}}$ to $C^{*}\mathbf{Cat}^{\mathrm{nu}}$,
 \item the tensor product $C_{0}(X)\otimes \mathbf{Q}_{\tau}(X)$,
 \item the multiplication morphism $\mu_{(X,\tau)} \colon C_{0}(X)\otimes \mathbf{Q}_{\tau}(X)\to \tilde \mathbf{Q}^{(G)}_{\mathrm{std}}$,
 \item the diagonal morphism $\delta_{(X,\tau)} \colon \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))\to \mathrm{KK}^{G}(C_{0}(X),C_{0}(X)\otimes \mathbf{Q}_{\tau}(X))$.
 \end{enumerate}

 We have a functor $$G\mathbf{UBC}^{\mathrm{scl}}\to G\mathbf{BC}\, , \quad (X,\tau)\mapsto {\mathcal{O}}_{\tau}(X)\otimes G_{can,max}\  ,$$
where ${\mathcal{O}}_{\tau}(X)$ is described in Definition \ref{qegoiejrgoeirgqwefqwfwqefqewf}.\ref{weigoggsfgsfgsfgsdfgsg}.
 Composing  with $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}$ from Definition \ref{wtogjoergrwegreggwege} we get a functor 
 \begin{equation}\label{rgfqfewfqewf1}
G\mathbf{UBC}^{\mathrm{scl}}\to {\mathbf{C}^{\ast}\mathbf{Cat}}\, ,\quad  (X,\tau)\mapsto \bD_{\tau}(X) \coloneqq   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(  {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})\, .
\end{equation} 
 We furthermore have the subfunctor  
 \begin{equation}\label{rgfqfewfqewf}
G\mathbf{UBC}^{\mathrm{scl}}\to C^{*}\mathbf{Cat}^{\mathrm{nu}}\, , \quad  (X,\tau)\mapsto  \bC_{\tau}(X) \coloneqq  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})
\end{equation}
(see \eqref{f23r8z89fz2893zf892334234f2}  {and \eqref{fqewqewfkqwpefowqefewfef}} for notation)
  such that $\bC_{\tau}(X)$ is a closed ideal in $\bD_{\tau}(X)$.
 Note that $\bC_{\tau}(X)$ is our replacement for
 $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( X\otimes G_{can,max})$ which can be considered as a subcategory of
 $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(  {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})$ of objects {which are} supported on $\{0\}\times X\times G$, but which is not an ideal {(these two $C^*$-categories actually have the same $K$-theory as will be used and also explained further below in Diagram \eqref{qfwefeeqfeqfqwfqewwfeqwf}).}
 Our choice of  notation  $ \bC_{\tau}(X)$ and $\bD_{\tau}(X)$ should indicate that
 these $C^{*}$-categories are our versions of the Roe algebra and the algebra of pseudolocal operators. We refer to Section \ref{weogiwjiergreggwrgwerg} for more details.  
 By forming quotients of $C^{*}$-categories we finally define the functor 
  \begin{equation}\label{qwefoujujfqew09ufewewfewfqfqef}
G\mathbf{UBC}^{\mathrm{scl}}\to C^{*}\mathbf{Cat}^{\mathrm{nu}}\,, \quad (X,\tau)\mapsto\mathbf{Q}_{\tau}(X) \coloneqq  \frac{\bD_{\tau}(X) }{  \bC_{\tau}(X)}\, .
\end{equation}
 Recall the  forgetful functor  $\iota^{\mathrm{scl}}\colon G\mathbf{UBC}^\mathrm{scl} \to G\mathbf{UBC}$, {the functor $K_{\bC}^{G,\cX}$ from Definition~\ref{qriofjqofewfefqeqff}, and 
  the $K$-theory functor $K^{C^{*}\mathrm{Cat}}$ for $C^{*}$-categories from \eqref{rewgwergwergegwergrewgregwregegwre}.}
 
  
 \begin{lem}\label{wrtoihgjoergergegwgrgwerg}
 We have a canonical  equivalence of functors
 \begin{equation}\label{regvoijoifevfdvafvfdsv}
K_{\bC}^{G,\cX}\circ \iota^{\mathrm{scl}}\simeq K^{C^{*}\mathrm{Cat}}\circ \mathbf{Q}_{\tau}\colon  G\mathbf{UBC}^{\mathrm{scl}} \to \Sp^{\mathrm{}}\, .
\end{equation} 
 \end{lem}
 \begin{proof}
  We have a  natural (naturality here and below  refers to   $(X,\tau)$ in $G\mathbf{UBC}^{\mathrm{scl}}$)   commuting diagram of  $C^{*}$-categories
\begin{equation}\label{qfqwfqewfeqedw}
\xymatrix{
&  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( X\otimes G_{can,max})
\ar[r]
\ar[d]&  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})\ar@{=}[d]
&
&
\\0\ar[r]&\bC_{\tau}(X)\ar[r]&\bD_{\tau}(X)\ar[r]&\mathbf{Q}_{\tau}(X)\ar[r]&0}
\end{equation}
where the {top} horizontal and left vertical morphisms are induced from canonical inclusions of bornological coarse spaces. 

We now use that $K^{C^{*}\mathrm{Cat}}$   sends sequences in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$ to fibre sequences in $\Sp^{\mathrm{}}$ (\cite[Thm.~1.29.5]{KKG} or \cite[Prop.\ 12.9]{cank}). We  apply $K^{C^{*}\mathrm{Cat}}$ to Diagram \eqref{qfqwfqewfeqedw} and get  a natural  morphism of fibre sequences
\begin{align}\label{qfwefeeqfeqfqwfqewwfeqwf}  
\\
\mathclap{\xymatrix{ 
 K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( X\otimes G_{can,max}))\ar[r]\ar[d]^{\simeq}&
K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}_{\tau}(X)\otimes G_{can,max}))\ar@{=}[d]\ar[r]&
P
\ar[d]^{\simeq}\\ 
K^{C^{*}\mathrm{Cat}} (\bC_{\tau}(X))\ar[r]&K^{C^{*}\mathrm{Cat}}(\bD_{\tau}(X))\ar[r]&K^{C^{*}\mathrm{Cat}}(\mathbf{Q}_{\tau}(X))
}}\notag 
\end{align}
in $\Sp^{\mathrm{}}$, where $P$ is  defined as the cofibre of the left upper horizontal morphism.
 The left vertical morphism is an equivalence by \cite[Lem.\ 5.6]{coarsek} (the main point is that the inclusions $X\otimes G_{can,max}\to Z_{n}$ for all $n$ in $\nat$ are coarse equivalences, where $Z_{n}$ is as in \eqref{fqewqewfkqwpefowqefewfef}).
It follows that the right vertical morphism is an equivalence, too.

We have a natural commuting diagram
$$\xymatrix{ K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( X\otimes G_{can,max})) \ar[r]\ar@{=}[d]&K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})) \ar[d]^{\simeq}\\     K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( X\otimes G_{can,max})) \ar[r]&     K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}} (X)\otimes G_{can,max}))}$$
where the right vertical morphism is given by the morphism $${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}\to {\mathcal{O}} (X)\otimes G_{can,max}$$ induced by the identity of the underlying spaces. It is an equivalence by Proposition~\ref{eqroguerogeregwergwre}
  since  the functor
$K\bC\cX^{G}_{c,G_{can,max}}(-)\simeq 
K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(-)\otimes G_{can,max}))$ is an equivariant  coarse homology theory.
This square extends to a natural morphism of fibre sequences
\begin{align}\label{qfwefeeqfeqfqwfqewwffeqwf}  
\\
\mathclap{ \xymatrix{ 
K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( X\otimes G_{can,max})) \ar[r]\ar@{=}[d]&K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}_{\tau}(X)\otimes G_{can,max})) \ar[d]^{\simeq}\ar[r] &\ar[d]^{\simeq}P
 \\     K\bC\cX_{c,G_{can,max}}^{G} ( X ) \ar[r]&  K\bC\cX_{c,G_{can,max}}^{G} ( {\mathcal{O}}(X) )\ar[r]&K\bC\cX_{c,G_{can,max}}^{G} ( {\mathcal{O}}^{\infty}(X) )
 }}\notag
 \end{align}
 where, by inserting definitions,  we have rewritten the lower sequence as an instance  of the cone sequence
   \eqref{wefqwefqewewfwdqewdewdqede} applied to $E \coloneqq K\bC\cX_{c,G_{can,max}}^{G}$.   
  Note that by Definition \ref{qriofjqofewfefqeqff} {we have}
  $K_{\bC}^{G,\cX}(X)\simeq  K\bC\cX_{c,G_{can,max}}^{G} ( {\mathcal{O}}^{\infty}(X) )$.
  Composing this equivalence with the inverse of the right vertical equivalence in 
 \eqref{qfwefeeqfeqfqwfqewwffeqwf} and the right vertical equivalence in \eqref{qfwefeeqfeqfqwfqewwfeqwf} yields the natural equivalence\begin{equation}\label{eqwfjoiwefqwfewfqewf}
K_{\bC}^{G,\cX}(X)\simeq K^{C^{*}\mathrm{Cat}}(\mathbf{Q}_{\tau}(X))\, .
\end{equation}
as desired.
\end{proof} 
  
  
  
  
 We now use the functor
 \begin{equation}\label{ewrgegwegvdsf}
\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}:\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\to \mathrm{KK}^{G}
\end{equation}
 from \cite[Def.~1.27]{KKG}, where we omit the supercripts $G$ in the case of the trivial group. We can furthermore omit the {subscript} $C^{*}{\mathbf{Cat}}$ if we apply this functor to $G$-$C^{*}$-algebras.
 Following the conventions in \cite{KKG} we will use the notation
 $$\mathrm{KK}^{G}(\bC,\bD):=\map_{\mathrm{KK}^{G}}(\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\bC),\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\bD))$$
 for $\bC,\bD$ in $ \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$.
  At various places we will use the following properties of this functor. Consider $Y$  in $G\mathbf{LCH}^{\mathrm{prop}}_{+}$.
\begin{lem}\label{eroigjweogregwgefwe}
If $Y$ is homotopy equivalent to a $G$-finite $G$-CW-complex with finite stabilizers, then:
\begin{enumerate}
\item\label{qroigwoeigwjereferwfwerf} $\mathrm{KK}^{G}(C_{0}(Y),-)$ sends exact sequences  in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ to fibre sequences.
\item\label{qroigwoeigwjereferwfwerf1}  $\mathrm{KK}^{G}(C_{0}(Y),-)$ annilhilates flasque objects  in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$.
\end{enumerate}
\end{lem}
\begin{proof}
By  \cite[Prop.~1.25]{KKG} the object $\mathrm{kk}^{G}(C_{0}(Y))$ is  ind-$G$-proper in the sense of   \cite[Prop.~1.24]{KKG}.
Both assertions now follow from  \cite[Thm.~1.29]{KKG}.
\end{proof}



 
 
 In the present paper $\otimes$ denotes the maximal tensor product of $C^{*}$-categories \cite[Def.~7.2]{KKG}. By \cite[Prop.~1.20]{KKG} the stable $\infty$-category category $\mathrm{KK}^{G}$ has a presentably symmetric monoidal 
 structure induced by the maximal tensor product of $C^{*}$-algebras, and by  \cite[Thm.~1.31]{KKG} the functor
  $\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ in \eqref{ewrgegwegvdsf} 
   has a symmetric monoidal refinement.   We define  the functor
 \begin{equation}\label{saCWDQKOIOJ1} 
 - \hatotimes - \colon
\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} ) \times    \mathrm{KK}\xrightarrow{\mathrm{kk}^{G}\times \mathrm{Res}^{\{1\}}_{G}}\mathrm{KK}^{G}\times \mathrm{KK}^{G}  \xrightarrow{\otimes} \mathrm{KK}^{G}\, ,
\end{equation}
where $\otimes $ is structure map of  the symmetric monoidal structure of $\mathrm{KK}^{G}$  
and $\mathrm{Res}^{\{1\}}_{G}$ is the restriction induced by the projection $G\to \{1\}$ from  \cite[Thm.~1.21]{KKG}  {(on $C^*$-algebras $\mathrm{Res}^{\{1\}}_{G}$ is given by equipping a $C^*$-algebra with the trivial $G$-action).}  
Using that  $\mathrm{KK}^{G}$ is   presentably symmetric monoidal  category and $\mathrm{Res}^{\{1\}}_{G}$ preserves
small colimits we see that   $\hatotimes$ preserves
small colimits in its second variable.   
    
    
Let   $    A$ be in $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Alg}} ) $ and $\mathbf{Q}$ be in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$.
\begin{lem}\label{glkbijowerfvvfevsdfvsfdv} 
 We have an  equivalence
 $$     A \hatotimes \mathrm{kk}_{C^{*}\mathbf{Cat}}(\mathbf{Q})\simeq  \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G} (    A\otimes \mathrm{Res}^{\{1\}}_{G}(\mathbf{Q}))$$
 which is natural in $A$ and $\mathbf{Q}$.
\end{lem}
\begin{proof}
The chain of natural equivalences
\begin{eqnarray*}
     A \hatotimes \mathrm{kk}_{C^{*}\mathbf{Cat}}( \mathbf{Q})&\stackrel{\text{def.}}{\simeq}&
\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(     A)    \otimes   \mathrm{Res}^{\{1\}}_{G}( \mathrm{kk}_{C^{*}\mathbf{Cat}}  (\mathbf{Q}))\\
&\stackrel{\scriptsize{\cite[\text{Thm.~1.21}]{KKG}}}{\simeq}&
\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}   ( A)    \otimes  \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G} (\mathrm{Res}^{\{1\}}_{G} (\mathbf{Q}))\\
&\stackrel{\scriptsize{\cite[\text{Thm.~1.31}]{KKG}}}{\simeq}&
 \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(  A\otimes \mathrm{Res}^{\{1\}}_{G} (\mathbf{Q}))
\end{eqnarray*}
gives the desired equivalence.
\end{proof}

From now on, in order to simplify the notation, we will write  $\mathbf{Q}$ instead of $\mathrm{Res}^{\{1\}}_{G} (\mathbf{Q})$.


 For $(X,\tau)$ in $G\mathbf{UBC}^{\mathrm{scl}}$ we have  {the objects} $C_{0}(X)$ in $\Fun(BG, {C^{*}\mathbf{Alg}^{\mathrm{nu}}})$ and 
    $\mathbf{Q}_{\tau}(X)$ in $ C^{*}\mathbf{Cat}^{\mathrm{nu}}$ and can thus define     $C_{0}(X)\otimes \mathbf{Q}_{\tau}(X)$ in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$, where  consider the left tensor factor as a $C^{*}$-category. The objects of this category are
    the objects of $ \mathbf{Q}_{\tau}(X)$, and the morphism spaces are certain completions of the algebraic tensor products of the 
 morphism spaces of $ \mathbf{Q}_{\tau}(X)$ with $C_{0}(X)$.
   
    Recall the Definition \ref{qeroighjoreqgreqfefwfqwef}.\ref{wrtogwprtgwergwergfreferf}   of $   \mathbf{Q}^{(G)}_{\mathrm{std}}$ in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$.
    We  define the multiplication  morphism     
  \begin{equation}\label{oreihoijfvoisfdvervfdsvdfvsvv}
\mu_{(X,\tau)} \colon C_{0}(X)\otimes \mathbf{Q}_{\tau}(X)\to    \mathbf{Q}^{(G)}_{\mathrm{std}}
\end{equation} 
  in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ as follows.
    \begin{enumerate} \item
    objects: The morphism $\mu_{(X,\tau)}$ sends the object $(C,\rho,\mu)$ to the object $(C,\rho)$ of
    $  \mathbf{Q}^{(G)}_{\mathrm{std}}$. 
    Note that 
    $(C,\rho)$ belongs to  $  \mathbf{Q}^{(G)}_{\mathrm{std}}$ since the underlying $G$-set of ${\mathcal{O}}_{\tau}(X)\otimes G_{can,max}$ is a free $G$-set.
    \item \label{vefvvdfvweewfvfdsv} morphisms: The morphism $\mu_{(X,\tau)}$ is defined on morphisms uniquely by the universal property of the maximal tensor product 
    of $C^{*}$-categories 
    such that it sends the morphism $f\otimes  [A]$    
    in $C_{0}(X)\otimes \mathbf{Q}_{\tau}(X)$ with 
    $A \colon (C',\rho',\mu')\to (C,\rho,\mu)$ to the morphism $[\phi(f)A]$ in $ \tilde \mathbf{Q}^{(G)}_{\mathrm{std}}$. Here the brackets {$[-]$} indicate  classes in the respective   quotients \eqref{qwefoujujfqew09ufewewfewfqfqef} and \eqref{wefwfwfwefflkjlqwepofp}, and $\phi(f)$ is defined in \eqref{ewqfpojlqkrmeflkwefqef}.
    
    To see that this map is well-defined note that if $A$ is in $\bC_{\tau}(X)$, then
    $\phi(f)A\in  {\mathbf{K}}^{(G)}_{\mathrm{std}}$ by Lemma \ref{ewiogjoerwgfdsfdgsfg}.
    Further, by  Lemma \ref{wtioghjwergfgrefwref} we have 
  $[\phi(f)A]=[A\phi'(f)]$ which  
    implies that
  this prescription is compatible with the composition and the involution.
        \end{enumerate}

  Finally, we define the diagonal morphism $\delta_{(X,\tau)}$ as the composition
 \begin{eqnarray}\label{rgergergegwegerg}
\delta_{(X,\tau)} \colon \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))&\simeq&\mathrm{KK}(\mathrm{kk}_{C^{*}\mathbf{Cat}}(\C),\mathrm{kk}_{C^{*}\mathbf{Cat}}(\mathbf{Q}_{\tau}(X))\\
&\stackrel{C_{0}(X) \hatotimes -}{\to}& \mathrm{KK}^{G}(C_{0}(X) \hatotimes\mathrm{kk}_{C^{*}\mathbf{Cat}}(\C),C_{0}(X) \hatotimes \mathrm{kk}_{C^{*}\mathbf{Cat}}(\mathbf{Q}_{\tau}(X)))\nonumber\\&\stackrel{!}{\simeq} & \mathrm{KK}^{G}(\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(C_{0}(X)\otimes \C) ,\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}( C_{0}(X) \otimes \mathbf{Q}_{\tau}(X)))\nonumber\\&\simeq& 
 \mathrm{KK}^{G}(C_{0}(X) ,C_{0}(X) \otimes \mathbf{Q}_{\tau}(X)) \, .\nonumber
\end{eqnarray}
The last  equivalence  is given by  the identification $C_{0}(X)\otimes \C\cong C_{0}(X)$, and the equivalence marked by $!$ uses Lemma  \ref{glkbijowerfvvfevsdfvsfdv}

 
Recall that $(X,\tau)$ is an object of   $G\mathbf{UBC}^{\mathrm{scl}}$. Recall also  the functors $\iota^{\mathrm{top}}$ and $\iota^{\mathrm{scl}}$ from {\eqref{q3oigfherqoifqrfvffqwefqeeqwfqeeqwfqefqefeeqfeqfewffefqewqewfqfqewf}.  } We now define the Paschke morphism whose existence was claimed in Theorem~\ref{qreoigjoergegqrgqerqfewf}.\ref{weiufhiqwefewwfqewfewf}.
 \begin{ddd}\label{qeorigjoqfeqwfqfewf}
The Paschke morphism {for $(X,\tau)$} is defined as the composition
\begin{align}\label{nlkkmlmvfdvsdva}
p_{(X,\tau)} \colon K_{\bC}^{G,\cX}(\iota^{\mathrm{scl}}(X,\tau)) \qquad & \mathclap{\stackrel{\eqref{regvoijoifevfdvafvfdsv}, \eqref{rewgwergwergegwergrewgregwregegwre}}{\simeq}}  \qquad \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))\\
& \mathclap{\stackrel{\delta_{(X,\tau)}}{\to}} \qquad \mathrm{KK}^{G}(C_{0}(X),C_{0}(X)\otimes \mathbf{Q}_{\tau}(X))\notag\\
& \mathclap{\stackrel{\mu_{(X,\tau)}}{\to}} \qquad \mathrm{KK}^{G}(C_{0}(X),   \mathbf{Q}^{(G)}_{\mathrm{std}})\notag\\
& \mathclap{\stackrel{\eqref{vfdsvsvvsvsdvdsadsvdsva}}{\simeq}} \qquad K_{\bC}^{G,\mathrm{An}}(\iota^{\mathrm{top}}(X,\tau))\,.\notag
\end{align}
\end{ddd}

Note that from this definition is not clear that the Paschke morphism is natural in $(X,\tau)$. The naturality will be discussed in the next Section  \ref{wtgiojioergerfwerfewfwerf}.


\section{Naturality of the Paschke morphism}\label{wtgiojioergerfwerfewfwerf}

In this subsection   {we} discuss the naturality of the Paschke morphism {from} Definition \ref{qeorigjoqfeqwfqfewf}.
More precisely, we will construct a natural transformation whose component on $(X,\tau)$ is the Paschke morphism of Definition \ref{qeorigjoqfeqwfqfewf}.
 Note that naturality in the $\infty$-categorical sense is more {than} the existence of a filler for the square 
\begin{equation}\label{fqwefoiwjfwoqefewfwqfe}
\xymatrix{K_{\bC}^{G,\cX}(\iota^{\mathrm{scl}}(X,\tau))\ar[r]^{f_{*}}\ar[d]^{p_{(X,\tau)}}&K_{\bC}^{G,\cX}(\iota^{\mathrm{scl}}(X',\tau'))\ar[d]^{p_{(X',\tau')}}\\ K_{\bC}^{G,\mathrm{An}}(\iota^{\mathrm{top}}(X,\tau))\ar[r]^{f_{*}}& K_{\bC}^{G,\mathrm{An}}(\iota^{\mathrm{top}}(X',\tau'))}
\end{equation}  
for all  morphisms $f \colon (X,\tau)\to (X',\tau')$ in $G\mathbf{UBC}^{\mathrm{scl}}$.
 The existence of such a filler  can indeed be easily seen by considering the big diagram \eqref{fqweflkfnlfewfqewfwfqfw} below.
 In  order to produce the data of a natural transformation we  must reformulate the construction 
 of the Paschke morphisms appropriately. 
 The main problem is that $\mathrm{KK}^{G}(C_{0}(X),C_{0}(X)\otimes \mathbf{Q}_{\tau}(X))$ is not a functor on $(X,\tau)$ so that  $\delta_{(X,\tau)}$ and $\mu_{(X,\tau)}$ can not be interpreted as natural transformations separately.

In order to get an idea  what we have to do  and get the existence of a filler of \eqref{fqwefoiwjfwoqefewfwqfe} we first consider the diagram
\begin{equation}\label{fqweflkfnlfewfqewfwfqfw}
\xymatrix{\mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))\ar[ddd]_{\mathrm{KK}^{G}(-,\mathbf{Q}(f))}\ar[r]^-{\delta_{(X,\tau)}}\ar@/_1.5pc/[ddr]^-(0.3){\delta_{(X',\tau')}}&\mathrm{KK}^{G}(C_{0}(X),C_{0}(X)\otimes \mathbf{Q}_{\tau}(X))\ar[d]^{\mathrm{KK}^{G}(f^{*},-)}\ar[r]^-{\mu_{(X,\tau)}}&\mathrm{KK}^{G}(C_{0}(X),  \mathbf{Q}_{\mathrm{std}}^{(G)})\ar[ddd]^{\mathrm{KK}^{G}(f^{*},-)}\\
&\mathrm{KK}^{G}(C_{0}(X'),C_{0}(X)\otimes \mathbf{Q}_{\tau}(X))\ar@/^1.5pc/[ddr]^-(0.7){\mu_{(X,\tau)}}&\\
&\mathrm{KK}^{G}(C_{0}(X'),C_{0}(X')\otimes \mathbf{Q}_{\tau}(X))\ar[u]_{\mathrm{KK}^{G}(-,f^{*})}\ar[d]^{\mathrm{KK}^{G}(-,\mathbf{Q}(f)}&\\
\mathrm{KK}(\C,\mathbf{Q}_{\tau'}(X'))\ar[r]^-{\delta_{(X',\tau')}}&\mathrm{KK}^{G}(C_{0}(X'),C_{0}(X')\otimes \mathbf{Q}_{\tau'}(X'))\ar[r]^-{\mu_{(X',\tau')}}&\mathrm{KK}^{G}(C_{0}(X'),  \mathbf{Q}_{\mathrm{std}}^{(G)})}
\end{equation}
all of whose  cells have essentially obvious fillers.  This  already  implies that the Paschke morphism is natural on the level of homotopy categories. 

\begin{rem}
Our idea for showing that the Paschke morphism is an equivalence is to reduce this by homotopy invariance to $G$-simplicial complexes, and then by excision to $G$-orbits where it can be verified by an explicit calculation. 
The excision step  requires a natural transformation on the spectrum level. 
If one is only interested in homotopy groups, then it would be sufficient to know the compatibility
of the Paschke map with the Mayer--Vietoris boundary maps which is an immediate consequence of the
spectrum-valued naturality.
So even if we were   finally  {only} interested in the Paschke isomorphism on the level of  homotopy groups
we would  {still} need the spectrum level natural transformation for the proof that it is an isomorphism.  

For similar reasons, the spectrum-valued version is also crucial in the proof of our second Theorem  \ref{wtoiguwegwergergregwe}
comparing the two assembly maps,  {though} the latter is indeed a statement on the level of homotopy groups.
\hspace*{\fill}$\qed$
\end{rem}
 


In the following remarks  about general $\infty$-categorical constructions we prepare the actual construction of the natural Paschke  transformation.

  \begin{rem} \label{wegijowergfwerfgwrf}For a category $\cC$ let $\mathbf{Tw}(\cC)$ denote the twisted arrow category.
 Objects are morphisms $C\to C'$ in $\cC$, and morphisms $(C_{0}\to C_{0}')\to (C_{1}\to C_{1}')$ are commuting diagrams 
 \begin{equation}\label{qewfoijoiqfewfqewfqewfqef}
\xymatrix{C_{0}\ar[r]&C_{0}'\ar[d]\\C_{1}\ar[r]\ar[u]&C_{1}'}
\end{equation}
 We have a canonical functor
 $$(\ev,\ev') \colon \mathbf{Tw}(\cC)\to \cC^{\mathrm{op}}\times \cC\, , \quad (C\to C')\mapsto (C,C')\, .$$
 If $F,G \colon \cC\to {\mathcal{D}}$ are two functors to a stable $\infty$-category, then we can express the spectrum of natural transformations 
 between $F$ and $G$ {as}
 \begin{equation}\label{wetliogjwoiergregregef}
\mathrm{nat}(F,G)\simeq \lim_{\mathbf{Tw}(\cC)} \map_{{\mathcal{D}}}(F\circ \ev,G\circ \ev')\, .
\end{equation}
 We refer to \cite{Gepner:2015aa,Glasman:2014aa} where this is discussed even  in the more general  case  of $\cC$ being an $\infty$-category.
 \hspace*{\fill}$\qed$
  \end{rem}
    
  
  
  \begin{rem}\label{qiuhqriegergrggregwergwegwerg}
  Recall that our universe in which we do homotopy theory is the one of small sets. 
  The corresponding categories then belong to the  large universe. 
  Let $\cC$ be a   locally small,  large presentable {stable} $\infty$-category and fix an object $C_{0}$ in $\cC$. 
  Then $C_{0}$ is $\kappa$-compact for some 
  small ordinal $\kappa$.
  The mapping space functor
  $\map_{\cC}(C_{0},-) \colon \cC\to \Sp^{\mathrm{}}$ preserves small limits and $\kappa$-filtered colimits. Consequently, it is the right adjoint in an adjunction 
  \begin{equation}\label{ewfoijqoiwejfoqwefqewqefef}
(E\mapsto C_{0}\wedge E):\Sp^{\mathrm{}}\leftrightarrows \cC:(C\mapsto \map_{\cC}(C_{0},C))\, .
\end{equation} 
  The left adjoint functor is essentially uniquely characterized by an equivalence  $C_{0}\simeq C_{0}\wedge S$, {where $S$ is the sphere spectrum,} and the fact that it preserves small colimits. This left adjoint  also depends functorially on $C_{0}$. 
  We  in fact have a functor
  \begin{equation}\label{sdvjqr0evj0fvsfvvfv}
\cC\times \Sp^{\mathrm{}}\to \cC\, ,\quad (C,E)\mapsto C\wedge E
\end{equation} 
  which is essentially uniquely characterized by a natural  equivalence \begin{equation}\label{qwfqewfqewfqewfqewffefefqewefefdfqef}
C\simeq C\wedge S
\end{equation}
and the fact that it preserves small  colimits in the second variable.
  
 The counit of the adjunction in \eqref{ewfoijqoiwejfoqwefqewqefef} is a natural transformation 
  \begin{equation}\label{qroujioqrfqefefeqwf}
    C_{0}\wedge \map_{\cC}(C_{0},-)\to \id_{\cC} (-) 
\end{equation} of endofunctors of $\cC$.
\hspace*{\fill}$\qed$
    \end{rem}
    
   
  \begin{rem}\label{weoigjwoigregwegwergrewgw}
  Let $\cC,{\mathcal{D}},{\mathcal{E}}$ be $\infty$-categories and $- \hatotimes - \colon \cC\times {\mathcal{D}}\to {\mathcal{E}}$ be a functor. We consider  $\infty$-categories ${\mathcal{I}}$, $\mathcal{J}$  and  
   natural transformations of functors $ {(F\stackrel{\alpha}{\to}  F')}\colon {\mathcal{I}}\to \cC$ and ${(G\stackrel{\beta}{\to} G')}\colon\mathcal{J}\to{\mathcal{D}}$. 
  Then we get a natural transformation of functors
  \[
  {(F\times G \stackrel{\alpha\times \beta}{\to} F'\times G')} \colon {\mathcal{I}}\times \mathcal{J}\to \cC\times {\mathcal{D}}\,,
  \]
  and by composition with $- \hatotimes -$ a natural transformation  \begin{equation}\label{fwrefwpofkwerpoffewrfwerf}
 {(F \hatotimes G \stackrel{\alpha \hatotimes \beta}{\to} F' \hatotimes G')} \colon {\mathcal{I}}\times \mathcal{J}\to {\mathcal{E}}\, ,
\end{equation}
 where we write
  $F \hatotimes G$ {for} $(- \hatotimes -)\circ (F\times G) $.
  \hspace*{\fill}$\qed$
    \end{rem}
  
 
 Applying    \eqref{sdvjqr0evj0fvsfvvfv}  to $\cC=\mathrm{KK}$ we get a functor 
 $$(B,E)\mapsto B\wedge E \colon \mathrm{KK}\times \Sp^{\mathrm{}}\to \mathrm{KK}\, .$$
 In the following we specialize $B$ to $\mathrm{kk}(\C)$.
 We then have a functor $ (A,E)\mapsto A\wedge E$  given as the composition  
 \begin{equation}\label{qewfpojfopqwfqewfeqf}
  \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\times \Sp^{\mathrm{}}\xrightarrow{\id\times {( \mathrm{kk}(\C)\wedge -)}} \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\times \mathrm{KK} \xrightarrow{- \hatotimes -} \mathrm{KK}^{G}\, ,
\end{equation}
where $\hat \otimes$ is as in \eqref{saCWDQKOIOJ1}.
Note that $$A\wedge S\stackrel{\eqref{qewfpojfopqwfqewfeqf}}{\simeq} A\hatotimes {(\mathrm{kk}(\C)\wedge S)}\stackrel{\eqref{qwfqewfqewfqewfqewffefefqewefefdfqef}}{\simeq}    A \hatotimes \mathrm{kk}_{C^{*}\mathbf{Cat}}(\C) \stackrel{Lem . \ref{glkbijowerfvvfevsdfvsfdv}}{\simeq} \mathrm{kk}^{G}(A\otimes \bC)\simeq \mathrm{kk}^{G}(A)\, ,$$
where in the intermediate steps we consider $\C$ as the tensor unit of $C^{*}\mathbf{Cat}^{\mathrm{nu}}$ in order to
apply Lemma  \ref{glkbijowerfvvfevsdfvsfdv} as stated.
Since the functor $-\hat \otimes-$ in  \eqref{saCWDQKOIOJ1} preserves small colimits in its second variable, the functor in \eqref{qewfpojfopqwfqewfeqf} is essentially uniquely determined by
the equivalence
$A\wedge S\simeq \mathrm{kk}^{G}(A)$ and the fact that it preserves small colimits in the second variable.
Furthermore, by the adjunction \eqref{ewfoijqoiwejfoqwefqewqefef} we have a natural equivalence 
\begin{equation}\label{wergoihweriogergrwfere}
\map_{\Sp^{\mathrm{}}}(E,\mathrm{KK}^{G}(A,B))\simeq \mathrm{KK}^{G}(A\wedge E,B)
\end{equation}
for $E$ in $\Sp^{\mathrm{}}$, $A$ in $\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$, and $B$ in $\mathrm{KK}^{G}$.

 
 
 
   We consider the functor 
\begin{equation}\label{ewfiuhqiowefqwefewfqewfw}
F \colon (G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\times \mathrm{KK}\xrightarrow{C_{0}(-)\times   \mathrm{KK}(\C,-)} \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}) \times \Sp^{\mathrm{}}\xrightarrow{ -\wedge -,\eqref{qewfpojfopqwfqewfeqf}} \mathrm{KK}^{G}
\end{equation}
written as 
$$((X,\tau),B)\mapsto C_{0}(X)\wedge     \mathrm{KK}(\C,B)\, .$$
We further consider the functor 
$$H \colon (G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\times \mathrm{KK}\xrightarrow{C_{0}(-)\times \id(-)}  \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}) \times \mathrm{KK} \xrightarrow{- \hatotimes -} \mathrm{KK}^{G}$$ written as
$$((X,\tau),B)\mapsto C_{0}(X) \hatotimes B\, .$$
 

We now  construct the diagonal
transformation 
\begin{equation}\label{qewfoijfiofwfwfqwfef}
{(F \stackrel{\tilde \delta}{\to} H)} \colon  (G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\times \mathrm{KK}\to \mathrm{KK}^{G}\, .
\end{equation}
 Its specialization at $(X,\tau)$ in $ G\mathbf{UBC}^{\mathrm{scl}}$ and $B$ in $\mathrm{KK}$ is a morphism 
 \begin{equation} 
\tilde \delta_{(X,\tau),B} \colon C_{0}(X) \wedge \mathrm{KK}(\C,B)\to C_{0}(X) \hatotimes B
\end{equation}
in $\mathrm{KK}^{G}$.  Inserting   \eqref{qewfpojfopqwfqewfeqf} into the definition \eqref{ewfiuhqiowefqwefewfqewfw} of $F$
we get 
 $$F=(- \hatotimes -)\circ (C_{0}(-)\times \mathrm{kk}(\C)\wedge   \mathrm{KK}(\C,-))\, .$$
We now obtain $\tilde \delta$ in \eqref{qewfoijfiofwfwfqwfef} by specializing \eqref{fwrefwpofkwerpoffewrfwerf} to the transformations
$$ {(C_{0}(-) \stackrel{\id}{\to} C_{0}(-))} \colon (G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\to  \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$$ 
and 
$$ (\mathrm{kk}(\C)\wedge   \mathrm{KK}(\C,-)\to  \id(-)) \colon \mathrm{KK}\to \mathrm{KK}$$  given by  \eqref{qroujioqrfqefefeqwf}.
 


 
We define the functor 
\begin{equation}\label{wregpkpwergwgreegergw}
Q \colon G\mathbf{UBC}^{\mathrm{scl}} \to \mathrm{KK}\, , \quad  (X,\tau)\mapsto Q_{\tau}(X) \coloneqq \mathrm{kk}_{C^{*}\mathbf{Cat}} (\mathbf{Q}_{\tau}(X))\,,
\end{equation}
 {see \eqref{qwefoujujfqew09ufewewfewfqfqef} for $\mathbf{Q}_\tau(X)$.} 
 Then we consider the functor
 \begin{equation}\label{wqefqwefwefqflkmklqf}
\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\to (G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\times \mathrm{KK}\, , \quad ((X,\tau)\to (X',\tau'))\mapsto (X',   Q_{\tau}(X))\, .
\end{equation}
 The pull-back of $\tilde \delta$ in \eqref{qewfoijfiofwfwfqwfef} along \eqref{wqefqwefwefqflkmklqf} yields 
 a natural transformation 
 \begin{equation}\label{roiegoijviojfeqwfqewfewfq}
 (\delta \colon C_{0}(-')\wedge \mathrm{KK}(\C,Q(-))\to  C_{0}(-') \hatotimes Q(-)):\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\to \mathrm{KK}^{G}
\end{equation} 
 whose evaluation at {an object} $f \colon (X,\tau)\to (X',\tau')$ in $\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}$ is a morphism
 \begin{equation}\label{ewqoifjeiqjfioqewfqewfwef}
  \delta_{f} \colon C_{0}(X')\wedge \mathrm{KK}(\C,Q_{\tau}(X))\to C_{0}(X') \hatotimes Q_{\tau}(X)
\end{equation}
in $\mathrm{KK}^{G}$.
 This is our version of the diagonal \eqref{rgergergegwegerg} as a natural transformation. In fact,  under the  canonical  equivalence
  \begin{align}\label{eq_identification_deltas_X_nat}
\mathrm{KK}^{G} & (C_{0}(X)\wedge \mathrm{KK}(\C,Q_{\tau}(X)), C_{0}( {X}) \hatotimes Q_{{\tau}}(X))\\
& \stackrel{\eqref{wergoihweriogergrwfere}}{\simeq} \map( \mathrm{KK}(\C,\mathbf{Q}_{{\tau}}(X)),\mathrm{KK}^{G}(C_{0}(X),C_{0}(X) \otimes \mathbf{Q}_{\tau}(X))\notag
\end{align}
the map $\delta_{\id_{(X,\tau)}}$ in \eqref{ewqoifjeiqjfioqewfqewfwef} corresponds to $\delta_{(X,\tau)}$ from \eqref{rgergergegwegerg}.   
 
 
 
We now construct a refinement \eqref{ewfiuhiu34gferqgfqfewfeqwfqe} of the family of multiplication maps $\mu_{(X,\tau)}$ from \eqref{oreihoijfvoisfdvervfdsvdfvsvv}  for all $(X,\tau)$ in 
$G\mathbf{UBC}^{\mathrm{scl}}$. We start with
 the functor
\begin{align*}
\mathclap{
C_{0}(-')\otimes \mathbf{Q}(-) \colon \mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\to  \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\, ,\quad  ((X,\tau)\to (X',\tau'))\mapsto C_{0}(X')\times  \mathbf{Q}_{\tau}(X)\,.
}
\end{align*}
We also consider $ \tilde \mathbf{Q}^{(G)}_{\mathrm{std}}
$ as a constant functor from $\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}$ to $  \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$.  As a start we construct 
  a natural transformation 
  \begin{equation}\label{frvjvoijoqvvvqwvwevqw}
(\tilde \mu \colon C_{0}(-')\otimes \mathbf{Q}(-)\to   \mathbf{Q}^{(G)}_{\mathrm{std}}) \colon \mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\to \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\, .
\end{equation} 
 For  every object $f \colon (X,\tau)\to ( X',\tau')$ in $\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}$ we must define a functor 
 \begin{equation}\label{eq_identification_mu_X_nat}
 \tilde \mu_{f} \colon C_{0}(X')\otimes \mathbf{Q}_{\tau}(X)\to   \mathbf{Q}^{(G)}_{\mathrm{std}}\, .
 \end{equation}
 This construction  extends the construction of ${\mu_{(X,\tau)}}$ in \eqref{oreihoijfvoisfdvervfdsvdfvsvv} which {will be} 
 recovered as $\mu_{(X,\tau)}=\tilde \mu_{\id_{(X,\tau)}}$.
 \begin{enumerate}
 \item objects: The functor $\tilde \mu_{f}$ sends the object $(C,\rho,\mu)$ in $C_{0}({X'})\otimes \mathbf{Q}_{\tau}(X)$ (hence an object of  $\mathbf{Q}_{\tau}(X)$) to the object $(C,\rho)$ in $   \mathbf{Q}^{(G)}_{\mathrm{std}}$.
 \item morphisms: If $[A] \colon (C',\rho',\mu')\to (C,\rho,\mu)$ is a morphism in $\mathbf{Q}_{\tau}(X)$ and $h$ is in $C_{0}({X'})$, then 
 $\tilde \mu_{f}(h\otimes [A]) \coloneqq [\phi(f^{*}h)A]${, see \eqref{ewqfpojlqkrmeflkwefqef} for the definition of $\phi$.}
 \end{enumerate}
The argument that the functor $\tilde \mu_{f}$ is well-defined is  the same  as for  ${\mu_{(X,\tau)}}$. 
We now check that $\tilde \mu \coloneqq ({\tilde{\mu}}_{f})_{f\in \mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}}$ is a natural transformation.
We consider a morphism $f\to g$ in $ \mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}$, see \eqref{qewfoijoiqfewfqewfqewfqef}. Since we work with the opposite of the twisted arrow category, it is given by a commutative diagram \begin{equation}\label{ewqfoijhqoiwefqwefewqfqewfef}
\xymatrix{(X,\tau)\ar[r]^{f}\ar[d]^{\alpha}&(X',\tau') \\(Y,\sigma)\ar[r]^{g}&(Y',\sigma')\ar[u]^{\beta}}
\end{equation}
 We must show that 
 $$\xymatrix{ C_{0}(X')\otimes \mathbf{Q}_{\tau}(X)\ar[rr]^{\beta^{*}\otimes \mathbf{Q}(\alpha)}\ar[dr]_{{\tilde{\mu}}_{f}}&&\ar[dl]^{{\tilde{\mu}}_{g}}C_{0}(Y')\otimes \mathbf{Q}_{\sigma}(Y) \\& \mathbf{Q}^{(G)}_{\mathrm{std}} &}$$
 commutes.
 \begin{enumerate}
 \item objects:  Let   $(C,\rho,\mu)$ be an object  in $C_{0}(X')\otimes \mathbf{Q}_{\tau}(X)$. Then we have the equality
   $${\tilde{\mu}}_{g}((\beta^{*}\otimes \mathbf{Q}(\alpha))(C,\rho,\mu))= {\tilde{\mu}}_{g}(C,\rho, {\alpha}_{*}\mu)=(C,\rho)= {\tilde{\mu}}_{f}(C,\rho,\mu)\, .$$
   \item morphisms:  Let $[A] \colon (C',\rho',\mu')\to (C,\rho,\mu)$ be  a morphism in $\mathbf{Q}_{\tau}(X)$ and  $h$ {be} in $C_{0}(X')$.
   Then we have the equality
   $$ {\tilde{\mu}}_{g}((\beta^{*}\otimes \mathbf{Q}(\alpha))( h\otimes [A]))= {\tilde{\mu}}_{g}(  \beta^{*}h\otimes [\alpha_{*}A]) =  {[\phi(g^*(\beta^*(h))) \alpha_*A]} = [(\alpha_{*}\phi)(g^{*}\beta^{*}h)A]\, .$$
   On the other hand,
   $$ {\tilde{\mu}}_{f}( h\otimes [A])=[\phi( {f^*}h)A]\, .$$
   The desired equality
   $$[\phi( {f^*} h)A]=[(\alpha_{*}\phi)( {g^*}\beta^{*}h)A]$$ now follows from the identity
  $$(\alpha_{*}\phi)(g^{*}\beta^{*}h)=\phi(\alpha^{*}g^{*}\beta^{*}h)=\phi(f^{*}h)$$ since
   $\alpha^{*}g^{*}\beta^{*}h=f^{*}h$ by the commutativity of \eqref{ewqfoijhqoiwefqwefewqfqewfef}.
 \end{enumerate}



 We post-compose the transformation in \eqref{frvjvoijoqvvvqwvwevqw} with the functor $\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ in \eqref{ewrgegwegvdsf}
  and get   a natural transformation
 \begin{equation}\label{r9u89fqfewewqfwefqwefqwefqwef}
(\mathrm{kk}^{G}(\tilde \mu) \colon \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(C_{0}(-')\otimes \mathbf{Q}(-))\to  Q^{(G)}_{\mathrm{std}}) \colon \mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\to\mathrm{KK}^{G}\, ,
\end{equation}
    where we use the  abbreviation 
 \[
  Q^{(G)}_{\mathrm{std}} \coloneqq \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}( \mathbf{Q}^{(G)}_{\mathrm{std}})\,.
 \]
 Composing the transformation   \eqref{r9u89fqfewewqfwefqwefqwefqwef}
 with the   equivalence $$C_{0}(-') \hatotimes     Q(-)\simeq \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(C_{0}(-')\otimes \mathbf{Q}(-))$$    given by Lemma \ref{glkbijowerfvvfevsdfvsfdv}  
 (see \eqref{wregpkpwergwgreegergw} for {the} notation $Q(-)$) we get a natural transformation   \begin{equation}\label{ewfiuhiu34gferqgfqfewfeqwfqe}
(  \mu \colon C_{0}(-') \hatotimes Q(-)\to \tilde Q^{(G)}_{\mathrm{std}}) \colon \mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}\to \mathrm{KK}^{G}\, .
\end{equation}    


The composition of \eqref{roiegoijviojfeqwfqewfewfq} and  \eqref{ewfiuhiu34gferqgfqfewfeqwfqe}
then gives a natural transformation
$$  (\mu\circ   \delta \colon C_{0}(-')\wedge \mathrm{KK}(\C,Q(-))\to {C_0(-')\otimes Q(-)} \to      Q^{(G)}_{\mathrm{std}}) \colon \mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}} \to \mathrm{KK}^{G}$$
 whose value at the object  $f \colon (X,\tau)\to (X',\tau')$ is the morphism 
$$\mu_{f}\circ \delta_{f} \colon C_{0}(X')\wedge \mathrm{KK}(\C,Q_{\tau}(X)) \to  {C_0(X') \otimes Q_{\tau}(X)} \to     Q^{(G)}_{\mathrm{std}}\, .$$
Equivalently, by \eqref{wetliogjwoiergregregef} and since the target functor is constant we can interpret 
this as a map   {of  spectra}
\begin{equation}\label{gpojeopkgwpegregrgergw}
S\to \mathrm{KK}^{G}(\colim_{\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}} C_{0}(-')\wedge \mathrm{KK}(\C,Q(-)),      Q^{(G)}_{\mathrm{std}})\, .
\end{equation}
Note that $\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}$ is small    and  the  presentable category $\mathrm{KK}^{G}$ admits all small colimits.
We now use the chain of canonical equivalences  
\begin{eqnarray*}\lefteqn{
 \mathrm{KK}^{G}(\colim_{\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})^{\mathrm{op}}} C_{0}(-')\wedge \mathrm{KK}(\C,Q(-)),      Q^{(G)}_{\mathrm{std}})}&&\\&\simeq& \lim_{\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})}
 \mathrm{KK}^{G}(  C_{0}(-')\wedge \mathrm{KK}(\C,Q(-)),      Q^{(G)}_{\mathrm{std}})\\&\stackrel{\eqref{wergoihweriogergrwfere}}{\simeq}& \lim_{\mathbf{Tw}(G\mathbf{UBC}^{\mathrm{scl}})}
 \map( \mathrm{KK}(\C,Q(-)) , \mathrm{KK}^{G}(  C_{0}(-') ,     Q^{(G)}_{\mathrm{std}}))
  \\
 &\stackrel{\eqref{wetliogjwoiergregregef}}{\simeq}&\mathrm{nat}(   \mathrm{KK}(\C,Q(-)),  \mathrm{KK}^{G}(  C_{0}({-'}) ,     Q^{(G)}_{\mathrm{std}}  ))\, ,
\end{eqnarray*}
where $\mathrm{nat}$ denotes the spectrum of natural transformations between functors from 
$G\mathbf{UBC}^{\mathrm{scl}}$ to $\Sp^{\mathrm{}}$.
Therefore \eqref{gpojeopkgwpegregrgergw} provides a map 
$$S\to \mathrm{nat}(   \mathrm{KK}(\C,Q(-)),  \mathrm{KK}^{G}(  C_{0}(-) ,     Q^{(G)}_{\mathrm{std}}  ))
\, .$$
 This is the desired natural transformation 
  \begin{equation}\label{ekjbjkbejkefeqwfqwefe}
p \colon \mathrm{KK}(\C,Q(-))\to  \mathrm{KK}^{G}(C_{0}(-), \tilde Q^{(G)}_{\mathrm{std}})
\end{equation} 
of functors from 
$G\mathbf{UBC}^{\mathrm{scl}}$ to $\Sp^{\mathrm{}}$.
It follows from the identifications of $\delta_{\id_{(X,\tau)}}$ {with $\delta_{(X,\tau)}$ by \eqref{eq_identification_deltas_X_nat}} and  {of} ${\tilde{\mu}}_{\id_{(X,\tau)}}$ {with $\mu_{(X,\tau)}$ stated after \eqref{eq_identification_mu_X_nat}} that the evaluation of $p$ at $X$ in $ G\mathbf{UBC}^{\mathrm{scl}}$ is equivalent to the morphism
$p_{(X,\tau)}$ from \eqref{nlkkmlmvfdvsdva}.


    Recall that we use the notation 
    $$ \mathrm{KK}(\C,Q_{\tau}(X))\simeq  \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))\simeq K_{\bC}^{G,\cX}(\iota^{\mathrm{scl}}(X,\tau)) \, , $$ and  $$  \mathrm{KK}^{G}(C_{0}(X),  Q^{(G)}_{\mathrm{std}})\simeq   \mathrm{KK}^{G}(C_{0}(X),  \mathbf{Q}^{(G)}_{\mathrm{std}})\simeq K_{\bC}^{G,\mathrm{An}}(\iota^{\mathrm{top}}(X,\tau))\, .$$
  Therefore \eqref{ekjbjkbejkefeqwfqwefe}  is the desired Paschke transformation
$$p \colon  K_{\bC}^{G,\cX}\circ \iota^{\mathrm{scl}} \to K_{\bC}^{G,\mathrm{An}}\circ \iota^{\mathrm{top}}\, .$$
This finishes the proof of  Theorem \ref{qreoigjoergegqrgqerqfewf}.\ref{weiufhiqwefewwfqewfewf}.


 \phantomsection \label{eroigheifogegegergewgerg}
\section{ Reduction to  \texorpdfstring{$\boldsymbol{G}$}{G}-orbits}\label{ewoijegoewrgrewfe}


In the section we reduce the verification of  Theorems \ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf} and   \ref{qreoigjoergegqrgqerqfewf}.\ref{wrtigjiogowrefwerfewrf} to the case of $G$-orbits.
A discrete $G$-uniform bornological coarse space is a $G$-set   with
  the minimal coarse and bornological structure and the discrete uniform structure.
An object  $Y$ of $G\Set$ can canonically be considered as a discrete object  in $G\mathbf{UBC}$ which we will also denote by~$Y$. Alternatively we may use the more informative, but lengthier notation $Y_{min,min,disc}$, where the first $min$ indicates the minimal coarse structure, the second $ min$ the minimal bornology, and finally $disc$ the discrete uniform structure.
 Note that the construction $Y\mapsto Y_{min,min,disc}$ is functorial only for maps between $G$-sets with finite fibres.


 

Let
$\phi \colon \R\to (0,\infty)$ be a bounded function such that $\phi_{|(-\infty,0]}\equiv 1$.
  If $Y$ is  in  $G\Set$, then we can consider the scale
$\tau_{disc} \coloneqq (\phi,\psi_{disc})$ on $Y$, where 
\begin{equation}\label{toijsorgbsfdv}
\psi_{disc}(t) \coloneqq \begin{cases} Y\times Y& \text{for } t\le 0\,, \\
\diag(Y)& \text{for } t>0\,. \end{cases}
\end{equation}

Recall the Definition \ref{qreoighwoefewqffqewffewfq} of $G\mathbf{UBC}^{\mathrm{scl}}$.
 \begin{ddd}
We say that $\phi$ is good if $(\{*\},\tau_{disc})$ belongs to $G\mathbf{UBC}^{\mathrm{scl}}$.
\end{ddd}

For example, the constant function $\phi\equiv 1$ is good.

Assume that $(X,\tau)$ is in $G\mathbf{UBC}^{\mathrm{scl}}$ with  
  $\tau=(\phi,\psi)$. Let furthermore $Y$ be in $G\Set$.
\begin{lem}\label{lem_discrete_good_scale}\mbox{}
\begin{enumerate}
\item
  If $X$ is not empty, then $\phi$ is good.
\item \label{item_discrete_good_scale_countable}
If $Y$ is countable and $\phi$ is good,   {then} $(Y ,\tau_{disc})$ belongs to $G\mathbf{UBC}^{\mathrm{scl}}$.
\end{enumerate}
\end{lem} 
\begin{proof} For the first assertion 
the only non-trivial condition to check is that $(*,(\phi,\tau_{disc}))$ satisfies the condition of Definition \ref{qreoighwoefewqffqewffewfq}.\ref{fhewquifhqwieufhiqwefqwef}.

Since $X$ is not empty we can choose a point $x$ in $X$ and consider the $G$-invariant subset  $Y \coloneqq Gx$. Then $Y$ with the structures induced from $X$ is a discrete object in $G\mathbf{UBC}$. By Lemma \ref{qeroighioerwfqweewfqffew} we have $(Y,\tau_{Y})\in G\mathbf{UBC}^{\mathrm{scl}}$, where $\tau_{Y}$ is the restriction of the scale $\tau$ to $Y$. The first entry of  
 $\tau_{Y}$ is still $\phi$.  Let $L$ be a $G$-invariant locally uniformly finite subset of ${\mathcal{O}}_{\tau_{Y}}(Y)\otimes G_{can,max}$. 
 Then one checks that the projection $\bar L \coloneqq \pr_{[0,\infty)\times G}(L)$ is a $G$-invariant locally uniformly finite subset of ${\mathcal{O}}_{(\phi,\psi_{disc})}(*)\otimes G_{can,max}$.
 
  
For the second assertion let $\bar L$ be a  $G$-invariant locally uniformly finite subset of 
${\mathcal{O}}_{\tau_{disc}}(*)\otimes G_{can,max}$. Then the preimage $L \coloneqq \pr^{-1}_{[0,\infty)\times G}(\bar L)$ is a  $G$-invariant locally uniformly finite subset of  $
 {\mathcal{O}}_{\tau_{disc}}(Y)\otimes G_{can,max}$.
\end{proof}





Let $\cF$ denote a family of subgroups of $G$. We will be mainly interested in the family $\mathbf{Fin}$ of finite subgroups{, but the next proposition works for any family $\cF$.} We let $G_{\cF}\Set$ be the category of very small $G$-sets with stabilizers in $\cF$. 

Let $(X,\tau)$ be in $G\mathbf{UBC}^{\mathrm{scl}}$.  We write $\tau=(\phi,\psi)$.
Below we use  the first entry $\phi$ in order to form $\tau_{disc}$.
 Recall the Definition \ref{qeorigjoqfeqwfqfewf} of the Paschke morphism.
\begin{prop}\label{eoirgjwergerregwgreg} 
Assume: 
\begin{enumerate}
\item \label{ergoiwejrgrgrgwrgwergw}
The Paschke morphism $p_{(S,\tau_{disc})}$ is an equivalence for every
 $S$ in $ G_{\cF}\mathbf{Orb}$.
\item \label{weliorgjwgwreegwre9} $X$ is homotopy equivalent to a  $G$-finite  $G$-simplicial complex with stabilizers in $\cF$  {and} with  structures induced by its spherical path metrics.
\end{enumerate}
Then the Paschke morphism for $(X,\tau)$ is an equivalence.
\end{prop}


\begin{proof}
The case of an empty $X$ is obvious. We therefore assume that $X$ is not empty  {and} argue by induction on the dimension  $n$ of the $G$-simplicial complex in  Assumption  {\ref{eoirgjwergerregwgreg}.}\ref{weliorgjwgwreegwre9}.  
We assume that the first component of all scales below is given by $\phi$ which is good by Lemma  \ref{lem_discrete_good_scale}.
 
 Assume that  $d=0$ and that  $(K,\tau_{K})$ is in $G\mathbf{UBC}^{\mathrm{scl}}$
 such that    $K$ is a $0$-dimensional  $G$-finite $G$-simplicial complex with stabilizers in $\cF$. Then by an inspection of the constructions we have  $p_{(K,\tau_{K})}\simeq p_{(K,\tau_{disc})}$.
 
 
 For every orbit  $S$ in $G\backslash K$  we consider the closed  invariant partition  $(S,K\setminus S)$ of $K$.
 Applying excision for the functors $K_{\bC}^{G,\cX}$ and $K_{\bC}^{G,\mathrm{An}}$  
we get the respective projections 
$q^{\cX}_{S } \colon K_{\bC}^{G,\cX}(K)\to K_{\bC}^{G,\cX}(S)$
and
$q^{\mathrm{an}}_{S} \colon K_{\bC}^{G,\mathrm{An}}(K)\to K_{\bC}^{G,\mathrm{An}}(S)$ for all $S$ in $G\backslash K$. 
We have a commuting square 
$$\xymatrix{K_{\bC}^{G,\cX}(K)\ar[rr]_-{\simeq}^-{\oplus_{S}q^{\cX}_{S}}\ar[d]^{p_{(K,\tau_{K})}}&&\ar[d]_{\simeq}^{\oplus_{S}p_{(S,\tau_{disc})}} \bigoplus_{S\in  G\backslash K}   K_{\bC}^{G,\cX}(S) \\
K_{\bC}^{G,\mathrm{An}}(K) \ar[rr]_-{\simeq}^-{\oplus_{S}q^{\mathrm{an}}_{S}}&&\bigoplus_{S\in   G\backslash K} K_{\bC}^{G,\mathrm{An}}(S)  }$$
where the horizontal morphisms are equivalences by excision, and the right vertical morphism is an equivalence by Assumption {\ref{eoirgjwergerregwgreg}.}\ref{ergoiwejrgrgrgwrgwergw}. Here it is important that the assumptions on  {$K$} imply that $G\backslash K$ is finite. 
Consequently, the left vertical morphism is an equivalence.
 
 
   
Let $n$ be in $\nat$ and    assume that we have shown  that $p_{(K,\tau_{K})} $ is an equivalence provided   $K$ is  $G$-finite  $G$-simplicial complex with stabilizers in $\cF$ of dimension $n$  {and} with  structures induced by its spherical path metrics.
Let then $(X,\tau)$ be in $G\mathbf{UBC}^{\mathrm{scl}}$ and 
assume that there exists a homotopy equivalence
$(X,\tau)\to (K,\tau_{K})$. By the naturality of the Paschke morphism 
  we can consider the  commutative square 
$$\xymatrix{K_{\bC}^{G,\cX}(X)\ar[r]^{\simeq}\ar[d]^{p_{(X,\tau)}}&K_{\bC}^{G,\cX}(K)\ar[d]_{\simeq}^{p_{(K,\tau_{K})}}\\ K_{\bC}^{G,\mathrm{An}}(X)\ar[r]^{\simeq}&K_{\bC}^{G,\mathrm{An}}(K)}$$
Because the functors $K_{\bC}^{G,\cX}$ and $K_{\bC}^{G,\mathrm{An}}$ are homotopy invariant by \cite[Thm.~1.14]{KKG}  and Proposition \ref{oirgjoerwgwegregw9}, respectively,    for  the notion of homotopy {from} Definition \ref{ergoijogiwregergergwergwegr},
the horizontal morphisms are equivalences. By assumption  the right vertical morphism in an equivalence, too. Consequently, the left vertical morphism is also an equivalence.
 

  




We now show the actual induction step.
Assume that $(K,\tau_{K})$ in $G\mathbf{UBC}^{\mathrm{scl}}$ is such that $K$ is a $G$-finite $G$-simplicial complex with stabilizers in $\cF$ of dimension $n$ with  structures induced by its spherical path metrics.
 Let $Y$ be the closed $1/2$-neighbourhood of the $(n-1)$-skeleton $K_{n-1}$ of $K$ and $Z$ be
 $K\setminus \mathrm{int}(Y)$.  Then $(Y,Z)$ is a closed decomposition of~$K$.
 
By  Lemma \ref{qeroighioerwfqweewfqffew} we can consider $Y$, $Z$ and $Y\cap Z$ as objects in $G\mathbf{UBC}^{\mathrm{scl}}$ with the scale induced from $(X,\tau)$.
We then have the following commutative diagram
\begin{equation}
\xymatrix{K_{\bC}^{G,\cX}(Y\cap Z)\ar[ddd]\ar[dr]_-{\simeq}^-{\quad p_{Y\cap Z}}\ar[rrr]& &&K_{\bC}^{G,\cX}(Z)\ar[ddd]\ar[dl]^-{\simeq}_-{p_{Z}}\\
&K_{\bC}^{G,\mathrm{An}}(Y\cap Z)\ar[r]\ar[d]&K_{\bC}^{G,\mathrm{An}}( Z)\ar[d]&\\
&K_{\bC}^{G,\mathrm{An}}(Y )\ar[r]&K_{\bC}^{G,\mathrm{An}}(K)&\\
K_{\bC}^{G,\cX}(Y)\ar[rrr]\ar[ur]^-{\simeq}_-{\ p_{Y}}&&&K_{\bC}^{G,\cX}(K)\ar[ul]^-{p_{(K,\tau_{K})}}}
\end{equation}
where we omitted the symbol for the scale at various places.
  Since $Y, Z$ and $Y\cap Z$ are homotopy equivalent in $G\mathbf{UBC}^{\mathrm{scl}}$ to $G$-finite $G$-simplicial complexes of dimension $<n$ with stabilizers in $\cF$  their Paschke morphisms are equivalences by {the} induction hypothesis. Since the functors $K_{\bC}^{G,\mathrm{An}}$ and $K_{\bC}^{G,\cX}$ 
 are excisive for  this closed decomposition {(for $K^{G,\mathrm{An}}_{\bC}$ we use \cite[Prop. 5.1.2]{KKG})} the inner and the outer square are push-out squares. 
 Altogether we can then conclude that the Paschke morphism $p_{(K,\tau_{K})}$ is an equivalence, too.
 \end{proof}


 
In order to prepare the proof of Theorem \ref{qreoigjoergegqrgqerqfewf}.\ref{wrtigjiogowrefwerfewrf}  we replace the Paschke morphism $p$ in Proposition \ref{eoirgjwergerregwgreg}  by the locally finite version $p^{\mathrm{lf}}$ with target $K^{G,\mathrm{An},\mathrm{lf}}_{\bC}$. In Assumption  \ref{eoirgjwergerregwgreg}.\ref{ergoiwejrgrgrgwrgwergw} we further replace $G_{\cF}\mathbf{Orb}$ by $G_{\cF}\Set$. {Note that this is a stronger assumption.  }
The argument for Proposition \ref{eoirgjwergerregwgreg}
then also shows the following statement. Let $(X,\tau)$ be in $G\mathbf{UBC}^{\mathrm{scl}}$. 
\begin{prop}\label{eoirgjwergerregwgreg-allg} 
Assume: 
\begin{enumerate}
\item \label{ergoiwejrgrgrgwrgwergw-allg}
The Paschke morphism ${p^{\mathrm{lf}}_{(S,\tau_{disc})}:K^{G,\cX}_{\bC}(S)\to K^{G,\mathrm{An},\mathrm{lf}}_{\bC}(S)}$ is an equivalence for every countable
 $S$ in $ G_{\cF}\Set$.
\item \label{weliorgjwgwreegwre9-allg} $X$ is homotopy equivalent to a countable,  finite-dimensional  $G$-simplicial complex with stabilizers in $\cF$  {and} with  structures induced by its spherical path metrics.
\end{enumerate}
Then the Paschke morphism ${p^{\mathrm{lf}}_{(X,\tau)}:K^{G,\cX}_{\bC}(X)\to K^{G,\mathrm{An},\mathrm{lf}}_{\bC }(X)}$ is an equivalence.
\end{prop}
\begin{proof}
Using the stronger Assumption  \ref{eoirgjwergerregwgreg-allg}.\ref{ergoiwejrgrgrgwrgwergw} instead of Assumption \ref{eoirgjwergerregwgreg}.\ref{ergoiwejrgrgrgwrgwergw} one can redo the proof of Proposition \ref{eoirgjwergerregwgreg} for {$ p^{\mathrm{lf}}$} avoiding the step where we decompose the zero-dimensional complex $K$ into a finite union of $G$-orbits
\end{proof}

In the following lemma we show that   Assumption   {\ref{eoirgjwergerregwgreg}.}\ref{ergoiwejrgrgrgwrgwergw} implies  Assumption  {\ref{eoirgjwergerregwgreg-allg}.}\ref{ergoiwejrgrgrgwrgwergw}
  provided $G$ is finite.
  
  \begin{lem} \label{wergoepgrwegefwf}We assume that $G$ is finite. If 
the Paschke morphism $p_{(T,\tau_{disc})}$ is an equivalence for every
 $T$ in $ G\mathbf{Orb}$, then  the Paschke morphism ${p^{\mathrm{lf}}_{(S,\tau_{disc})}}$ is an equivalence for every countable
 $S$ in $ G\Set$.
\end{lem}
\begin{proof}

 {The functor 
 $K^{G,\mathrm{An},\mathrm{lf}}_{\bC}$}  sends countable disjoint unions  into
products.
Hence we have an equivalence
\begin{equation}\label{qewfqojkdjsd}
K^{G,\mathrm{An}{,\mathrm{lf}}}_{\bC}(S_{disc})\simeq \prod_{T\in G\backslash S} K^{G,\mathrm{An}{,\mathrm{lf}}}_{\bC}(T_{disc})\, .
\end{equation}
   If $G$ is finite, then we have an equality $G_{can,max}=G_{max,max}$. Recall the notion of the free union from \cite[Ex.~2.16]{equicoarse}.
As in the proof 
   of \cite[Lem.\ 3.13]{equicoarse}, by exploiting the equality $G_{can,max}=G_{max,max}$,  we have an isomorphism
\begin{equation}\label{ccdscqcasdcascasdca}
S_{min,min}\otimes G_{can, {max}}\cong (\bigsqcup^{\mathrm{free}}_{T\in G\backslash S} T_{min,min}) \otimes G_{can,max}\cong 
\bigsqcup_{T\in G\backslash S}^{\mathrm{free}} (T_{min,min} \otimes G_{can,max})\, .
\end{equation}
in $G\mathbf{BC}$.
  Since $K\bC\cX^{G}_{c}$ is strongly additive by
\cite[Thm.~11.1]{coarsek} it sends free unions to products. Applying now $K\bC\cX^{G}_{c}$ to \eqref{ccdscqcasdcascasdca} and
  using Definition \ref{qriofjqofewfefqeqff}  we consequently   have an equivalence 
\begin{equation}\label{qewfqojkdjsd1}
K^{G,\cX}_{\bC}(S_{min,min,disc})\simeq \prod_{T\in G\backslash S} K^{G,\mathrm{An}{,\mathrm{lf}}}_{\bC}(T_{min,min,disc})
\end{equation}
 {arising in the following way:
\begin{align*}
K^{G,\cX}_{\bC}(S_{min,min,disc}) \ & = \ K\bC\cX^{G}_{c,G_{can,max}}( {\mathcal{O}}^\infty ( S_{min,min,disc} ) )\\
&\stackrel{{!}}{ \simeq }\ \Sigma K\bC\cX^{G}_{c,G_{can,max}}( S_{min,min,disc} )\\
& \stackrel{\mathclap{\eqref{ccdscqcasdcascasdca}}}\simeq \ \Sigma K\bC\cX^{G}_{c} \Big( \bigsqcup_{T\in G\backslash S}^{\mathrm{free}} (T_{min,min} \otimes G_{can,max}) \Big)\\
& \simeq \ \Sigma \prod_{T\in G\backslash S} K\bC\cX^{G}_{c} ( T_{min,min} \otimes G_{can,max} )\\
& \simeq \ \prod_{T\in G\backslash S} K^{G,\cX}_{\bC} ( T_{min,min,disc} )\\
& \stackrel{\mathclap{p_{(T,\tau_{disc})}}}\simeq \ \prod_{T\in G\backslash S} K^{G,\mathrm{An}}_{\bC}(T_{min,min,disc})\\   &\simeq \ \prod_{T\in G\backslash S} K^{G,\mathrm{An}{,\mathrm{lf}}}_{\bC}(T_{min,min,disc})
\end{align*}
{Here we use \cite[Prop.\ 9.35]{equicoarse} for the  equivalence  marked by $!$.}
By} naturality of the Paschke transformation,  under the equivalences \eqref{qewfqojkdjsd} and   \eqref{qewfqojkdjsd1} the Paschke morphism $p^{{\mathrm{lf}}}_{(S,\tau_{disc})}$ corresponds to the product of the Paschke morphisms $p_{(T,\tau_{disc})}$ for the $G$-orbits $T$  in $S$. 
If the latter are equivalences, then $p_{(S,\tau_{disc})}$ is an equivalence.
\end{proof}

At the moment we do not know whether this lemma generalizes to infinite groups, possibly with restrictions on allowed stabilizers.


 
Combining Proposition \ref{eoirgjwergerregwgreg-allg} with Lemma \ref{wergoepgrwegefwf} we get the following result.
\begin{kor}\label{wtoigjwgowergwregwregw} 
Assume: 
\begin{enumerate}
\item $G$ is finite.
\item \label{ergoiwejrgrgrgwrgwergw-allg1} The Paschke morphism $p_{(T,\tau_{disc})}$ is an equivalence for every
 $T$ in $ G\mathbf{Orb}$.
 \item \label{weliorgjwgwreegwre9-allg1} $X$ is homotopy equivalent to a countable,  finite-dimensional  $G$-simplicial complex  with  structures induced by its spherical path metrics.
\end{enumerate}
Then the Paschke morphism {$p^{\mathrm{lf}}_{(X,\tau)}:K^{G,\cX}_{\bC}(X)\to K^{G,\mathrm{An},\mathrm{lf}}_{\bC}(X)$}   is an equivalence
\end{kor}

  


\begin{rem}\label{aergijoarvvavadsvsdva}
We can not expect that the Paschke morphism is an equivalence for spaces which are not proper $G$-spaces. More precisely, we do not expect that  Assumption \ref{eoirgjwergerregwgreg}.\ref{ergoiwejrgrgrgwrgwergw}    is  satisfied if $\cF$ contains infinite subgroups.

Assume that $S =G/H$ with $H$ infinite. 
Then we have  \begin{eqnarray}
K_{\bC}^{G,\cX}((G/H)_{min,min,disc})&\stackrel{\text{def.}}{\simeq} &K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}((G/H)_{min,min,disc} )))\nonumber\\&\stackrel{(1)}{\simeq}&\Sigma K\bC\cX^{G}_{c,G_{can,max}}((G/H)_{min,min})
\label{rgpfojewropgegwgwrgrgwegwerg}\\
&\stackrel{\text{def.}}{\simeq}&
 \Sigma K\bC\cX^{G}_{c}( (G/H)_{min,min}\otimes G_{can,max})\nonumber\\&\stackrel{(2)}{\simeq} &
 0\, , \nonumber
\end{eqnarray}
 {where t}he equivalence $(1)$ is an instance of \cite[Prop.\ 9.35]{equicoarse} since $(G/H)_{min,min,disc}$ is discrete. 
In order to see the equivalence $(2)$  we use that the functor   $K\bC\cX^{G}_{c}$ is continuous: We refer  
 to \cite[Def.\ 5.15]{equicoarse} for the definition of this notion and to \cite[Thm.\ 6.3]{coarsek} for the fact.
Continuity implies that the value of $K\bC\cX^{G}_{c}(X)$ for any $X$ in $G\mathbf{BC}$  is given as a colimit  of the values
$K\bC\cX^{G}_{c}(L)$ over the locally finite invariant subsets $L$ of $X$. 
We now observe that  if $H$ is infinite, then
$(G/H)_{min,min}\otimes G_{can,max}$ does not admit any non-empty invariant locally finite subset. 
Indeed, if $L$ would be such a subset, then on the one hand 
$(eH\times G)\cap L$ is finite,  but the infinite group $H$ acts freely on this set on the other hand.


 In contrast,  the spectrum
$$K_{\bC}^{G,\mathrm{An}} ( (G/H)_{disc})\simeq \mathrm{KK}^{G}(C_{0}((G/H)_{disc}),  \mathbf{Q}^{(G)}_{\mathrm{std}})$$ does not vanish in general.
As an example we consider the case $G=H$, and we  further specialize to 
$\bC=\mathbf{Hilb}^{G}(A)$ and ${\mathbf{K}}=\mathbf{Hilb}^{G}_{c}(A)$ for a $G$-$C^{*}$-algebra $A$.
 By Proposition \ref{wegojeogrregregwergwegre}
 we have an equivalence 
$$K_{\bC}^{G,\mathrm{An}} ( (G/H)_{disc})  \simeq \Sigma  \mathrm{KK}^{G}(\C,A)\, .$$
We claim that this spectrum is non-trivial if we take $A=\C$ with the trivial $G$-action.
Indeed, in this case we have the class $ \id_{\mathrm{kk}^{G}(\C)}$ in $\mathrm{KK}^{G}_{0}(\C,\C)$ and
$ \id_{\mathrm{kk}^{G}(\C)}\simeq 0$ if and only if $\mathrm{KK}^{G}(\C,\C)\simeq 0$.
 Since $\mathrm{kk}^{G}(\C)$ is the  tensor  unit of $\mathrm{KK}^{G}$
 we have  $  \mathrm{KK}^{G}(\C,\C)\simeq 0$ if and only if $\mathrm{KK}^{G}\simeq 0$. 
 But since
 $$\mathrm{KK}^{G}(C_{0}(G),\C)\simeq K^{C^{*}\mathrm{Alg}}(\C)\simeq KU$$ by \cite[Cor.\ 1.23]{KKG}
this never happens.
\hspace*{\fill}$\qed$
\end{rem}
 
 

  
Let $(X,\tau)$ be in $G\mathbf{UBC}^{\mathrm{scl}}$.  Then we have the multiplication map \eqref{oreihoijfvoisfdvervfdsvdfvsvv}
$$\mu^{\mathbf{Q}}_{X} \colon C_{0}(X)\otimes \mathbf{Q}_{\tau}(X)\to \tilde \mathbf{Q}^{(G)}_{\mathrm{std}}\, .$$
We add a superscript $\mathbf{Q}$ since we are going to consider other versions of this map which will be distinguished by other choices for this superscript. We further omitted the scale $\tau$ from the notation in order to simplify.
The main ingredient in the verification that $\mu^{\mathbf{Q}}_{X}$ is well-defined was
 Lemma  \ref{wtioghjwergfgrefwref}
 saying that for a morphism 
  $A \colon (C,\rho,\mu)\to (C',\rho',\mu')$ in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}_{\tau}(X)\otimes G_{can,max})$
we have $\phi'(f)A-A\phi(f)\in {\mathbf{K}}$  for all  $f$ in $C_{0}(X)$.
If $X$ is discrete, 
then we actually have
$\phi'(f)A-A\phi(f)=0$ for all  such $f$.
This has the effect that  in the construction of $\mu_{(X,\tau)}$  {in \eqref{oreihoijfvoisfdvervfdsvdfvsvv}} on morphisms  {(see Item  \ref{vefvvdfvweewfvfdsv} in the list below   \eqref{oreihoijfvoisfdvervfdsvdfvsvv})} we do not have to go to the quotients in order to ensure compatibility with the composition.
 

From now one we assume that $(X,\tau)$  is discrete.   
Using the observation just made  we can lift $\mu^{\mathbf{Q}}_{X}$ to a multiplication map
$$\mu^{\bD}_{X} \colon C_{0}(X)\otimes  \bD_{\tau}(X)\to     \bC^{(G)}_{\mathrm{std}}\, , \quad    f\otimes A\mapsto fA\, ,$$ where    $\bD_{\tau}(X)$ is defined in \eqref{rgfqfewfqewf1} 
Using in addition  Lemma  \ref{wtioghjwergfgrefwref}  and the definition \eqref{rgfqfewfqewf}  of $\bC_{\tau}(X)$
the map $\mu^{\bD}_{X} $ restricts to a map $\mu^{\bC}_{X}$ so that 
 we get a morphism of exact sequences in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ 
\begin{equation}\label{qfqwfqewfeqrrredw}
\xymatrix{0\ar[r]& C_{0}(X)\otimes \bC_{\tau}(X)\ar[r]\ar[d]^{\mu_{X}^{\bC}}& C_{0}(X)\otimes \bD_{\tau}(X)\ar[d]^{\mu_{X}^{\bD}} \ar[r]& C_{0}(X)\otimes \mathbf{Q}_{\tau}(X)\ar[d]^{\mu_{X}^{\mathbf{Q}}}\ar[r]&0\\0\ar[r]& {\mathbf{K}}^{(G)}_{\mathrm{std}}\ar[r]& \bC^{(G)}_{\mathrm{std}}\ar[r]& \mathbf{Q}^{(G)}_{\mathrm{std}}\ar[r]&0}
\end{equation}
 Here in the upper line  we used \eqref{qwefoujujfqew09ufewewfewfqfqef} and  that $C_{0}(X)\otimes -$  preserves exact sequences  of $C^{*}$-categories by  
 the exactness of the maximal tensor product for $C^{*}$-categories \cite[Prop. 7.21]{KKG}.


In the definition \eqref{rgergergegwegerg} of  the diagonal morphism $ \delta_{(X,\tau)}$
we could replace $\mathbf{Q}_{\tau}(X)$ by $\bC_{\tau}(X)$ or $\bD_{\tau}(X)$. 
Using the obvious   naturality of the construction of $\delta_{(X,\tau)}$ in this  variable
we get a commutative diagram \begin{align}\label{qfwefeeqfeqfqwfqerrrwwfeqwf}
\\
\mathclap{
\xymatrix{ 
 \mathrm{KK}(\C,\bC_{\tau}(X))\ar[r]\ar[d]^{\delta_{X}^{\bC}} & \mathrm{KK}(\C,\bD_{\tau}(X))
 \ar[d]^{\delta_{X}^{\bD}}\ar[r]& \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))
 \ar[d]^{\delta_{X}^{\mathbf{Q}}} 
 \\ \mathrm{KK}^{G}(C_{0}(X),C_{0}(X)\otimes \bC_{\tau}(X))  \ar[r]& \mathrm{KK}^{G}(C_{0}(X),C_{0}(X)\otimes \bD_{\tau}(X)) \ar[r]& \mathrm{KK}^{G}(C_{0}(X),C_{0}(X)\otimes \mathbf{Q}_{\tau}(X))  
 }}\notag\ .
\end{align}
Recall that we assume that $X$ is discrete. We now in addition assume that $X$ is $G$-finite and has finite stabilizers.
Using the exactness of the upper horizontal sequence in \eqref{qfqwfqewfeqrrredw} and  \eqref{qwefoujujfqew09ufewewfewfqfqef} 
we can conclude with Lemma  \ref{eroigjweogregwgefwe}.\ref{qroigwoeigwjereferwfwerf}
 that the horizontal sequences are segments of fibre sequences.
 Applying $\mathrm{KK}^{G}(C_{0}(X),-)$ to   \eqref{qfqwfqewfeqrrredw} and composing  the resulting   morphism of fibre sequences with the morphism   \eqref{qfwefeeqfeqfqwfqerrrwwfeqwf} 
we get the morphism of fibre sequences
\begin{equation}\label{qfwefeeqfeqfqwfqffferrrwwfeqwf}  
\xymatrix{ 
 \mathrm{KK}(\C,\bC_{\tau}(X))\ar[r]\ar[d]^{p_{X}^{\bC}} & \mathrm{KK}(\C,\bD_{\tau}(X))
 \ar[d]^{p^{\bD}_{X}}\ar[r]& \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))
 \ar[d]^{p_{X}^{\mathbf{Q}}}
 \\ \mathrm{KK}^{G}(C_{0}(X), {\mathbf{K}}^{(G)}_{\mathrm{std}})\ar[r]&\mathrm{KK}^{G}(C_{0}(X), \bC^{(G)}_{\mathrm{std}})\ar[r]& \mathrm{KK}^{G}(C_{0}(X),  \mathbf{Q}^{(G)}_{\mathrm{std}}) }
\end{equation}
where $p_{X}^{\mathbf{Q}}$ is the Paschke morphism \eqref{nlkkmlmvfdvsdva}.

For a family of subgroups $\cF$ we denote by $G_{\cF}\Set$   the full subcategory of $G\Set$ of $G$-sets with   stabilizers in $\cF$.
Let $(Y,\tau)$ be  a discrete object  of $G\mathbf{UBC}^{\mathrm{scl}}$.
\begin{lem}\label{lem_pseudoloc_vanishes_discrete}\mbox{}
\begin{enumerate}
\item
We have $ \mathrm{KK}(\C,\bD_{\tau}(Y))\simeq 0$.
\item \label{erthojrehtrhtrgegrge}
If $Y$ is in $G_{\mathbf{Fin}}\Set$ 
and $G\backslash Y$ is finite,   then   $\mathrm{KK}^{G}(C_{0}(Y),\bC^{(G)}_{\mathrm{std}})\simeq 0$.
\end{enumerate}
\end{lem}
\begin{proof}
We have the chain of equivalences: 
\begin{eqnarray*}
 \mathrm{KK}(\C,\bD_{\tau}(Y))&\stackrel{\eqref{rgfqfewfqewf1} \ \& \  \text{Def.~}\ref{qrogijeqoifefewfefewqffe}}{\simeq}&
 K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}_{\tau}(Y))\\
 &\stackrel{\text{Prop.~}\ref{eqroguerogeregwergwre}}{\simeq}& K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}(Y))\\&\simeq&
 0
\end{eqnarray*}
since  the cone ${\mathcal{O}}(Y)$  of a discrete object in $G\mathbf{UBC}$ is a flasque object in $G\mathbf{BC}$ by \cite[Ex. 9.25]{equicoarse}  and
the coarse homology theory  $K\bC\cX^{G}_{c,G_{can,max}}$ vanishes on flasques.


 

Recall that by our standing Assumption \ref{wergoiewgewrergwrgwgwg}  the $C^{*}$-category $\bC$ is  countably additive. 
This implies that
$  \bC^{(G)}_{\mathrm{std}}$ is  countably additive, too. 
Therefore it is flasque by \cite[Ex.\ {9.4}]{cank}. 
By Lemma   \ref{eroigjweogregwgefwe}.\ref{qroigwoeigwjereferwfwerf1}
we conclude that
$\mathrm{KK}^{G}(C_{0}(Y), \bC^{(G)}_{\mathrm{std}})\simeq 0$.
\end{proof}

Using Lemma \ref{lem_pseudoloc_vanishes_discrete} and  the morphism of fibre sequences \eqref{qfwefeeqfeqfqwfqffferrrwwfeqwf}
we get the following corollary.
\begin{kor}\label{qerpgoijqpwfeqfewfq}
If $X$ is in $G_{\mathbf{Fin}}\mathbf{Orb}$, then we have a commutative square $$\xymatrix{\Omega \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))\ar[r]^{\simeq}\ar[d]^{\Omega p^{\mathbf{Q}}_{X}}& \mathrm{KK}(\C,\bC_{\tau}(X))\ar[d]^{p_{X}^{\bC}}\\ \Omega  \mathrm{KK}^{G}(C_{0}(X),\mathbf{Q}^{(G)}_{\mathrm{std}})\ar[r]^{\simeq}& \mathrm{KK}^{G}(C_{0}(X), {\mathbf{K}}^{(G)}_{\mathrm{std}})}$$
In particular, the Paschke morphism for $X$ in $G_{\mathbf{Fin}} \mathbf{Orb}$ is an equivalence if and only if
the morphism $p_{X}^{\bC} \coloneqq \mu_{X}^{\bC}\circ \delta_{X}^{\bC}$ is an equivalence.
\end{kor}

In view of Corollary \ref{qerpgoijqpwfeqfewfq} and Proposition \ref{eoirgjwergerregwgreg} and  {Corollary  \ref{wtoigjwgowergwregwregw}} the following proposition finishes the proof of 
Theorem \ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf} and  \ref{qreoigjoergegqrgqerqfewf}.\ref{wrtigjiogowrefwerfewrf}:
 \begin{prop}\label{qeoigjqergqwedewdqed}
 Assume:
\begin{enumerate}
\item  
The ideal ${\mathbf{K}}$ is compatible with orthogonal sums {(see Definition \ref{wefiqrjhfoifewqdqwedqwedqewd}).}
\item $X$ is in $G_{\mathbf{Fin}}\mathbf{Orb}$. 
\end{enumerate}
Then 
\begin{equation}\label{verwejvweopvwerverwv}
p_{X}^{\bC} \colon \mathrm{KK}(\C,\bC_{\tau}(X))\to  \mathrm{KK}^{G}(C_{0}(X),{\mathbf{K}}^{(G)}_{\mathrm{std}})
\end{equation} 
is an equivalence.
\end{prop}

The whole Section \ref{oifjqweofwefewffqfef} is devoted to the proof of this proposition.

 
 
\section{Verification of the Paschke equivalence on \texorpdfstring{$\boldsymbol{G}$}{G}-orbits}\label{oifjqweofwefewffqfef}

 

 

We fix a finite subgroup  $H$   of $G$ and consider the $G$-set  $G/H$ in $G_{\mathbf{Fin}}\mathbf{Orb}$. 
As a first step we construct an explicit functor $\Theta$ in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$ and show in Proposition \ref{wqeroighowergergregrgwergwreg} that
$p^{\bC}_{G/H}$ is an equivalence if and only if $K^{C^{*}\mathrm{Cat}}(\Theta)$ is an equivalence.
In the second  step we then verify in Proposition \ref{4oijotrherhrthrhe} that $K^{C^{*}\mathrm{Cat}}(\Theta)$ is an equivalence provided that
${\mathbf{K}}$ is compatible with direct sums.


We form
 the $G$-bornological coarse space
$(G/H)_{min,min}\otimes G_{can,max}$.
It contains  
  the locally finite subset   
\begin{equation}\label{qrewfpokfpowefqweffqwfwefq}
X \coloneqq G(H, e)\, ,
\end{equation}
the $G$-orbit of the point $(H,e)$ in $G/H\times G$.  
Note that, in contrast to the example in Remark \ref{aergijoarvvavadsvsdva}, here the group $H$ is finite.
We equip $X$
 with the  bornological coarse structures  induced from $(G/H)_{min,min}\otimes G_{can,max}$.    Note that the underlying $G$-set of $X$ is isomorphic to $G$, and 
 the  induced bornology on $X$ is the minimal one. The induced $G$-coarse structure
 reflects the information about the finite subgroup $H$ and is in general smaller than the canonical coarse structure on $G$.
 
 
 
  The following lemma states 
  that the inclusion  $X\to  (G/H)_{min,min}\otimes G_{can,max}$
  is a continuous equivalence in the sense of   
\cite[Sec.~7]{desc}. 
 \begin{lem}\label{wegoijwegowiergwegerw}
 The inclusion $X\to  (G/H)_{min,min}\otimes G_{can,max}$ induces an equivalence
 $E(X)\to E((G/H)_{min,min}\otimes G_{can,max})$  for any continuous equivariant homology theory $E$.
 \end{lem}
 \begin{proof} 
For $Y$ in $G\mathbf{BC}$ we let $\mathrm{LF}(Y)$ denote the poset of $G$-invariant locally finite subsets.
Let $L$ be  in $\mathrm{LF}((G/H)_{min,min}\otimes G_{can,max})$.
Then $L_{0} \coloneqq L\cap ( \{H\}\times G)  $ is a finite set which we will sometimes consider as a subset of $G$. Since every $G$-orbit in $L$ meets $L_{0}$ we have $L = GL_{0}$.
  
  We claim that 
  for every $L$ in $\mathrm{LF}((G/H)_{min,min}\otimes G_{can,max})$  the inclusion  $i \colon X\to L\cup X$ is a coarse equivalence.
 Indeed, we can construct
 an  inverse equivalence $p: L\cup X\to X$. The 
 map $p$ is the identity on $X$, and  it sends a point $g(H,h)$ (with $h$ in $L_{0}\setminus \{e\}$) in $L\setminus X$ to $g(H,e)$ in $X$. Then $p\circ i=\id_{X}$
 and  $i\circ p $ is close to the identity. In order to see the second assertion note that $L_{0}$ is finite and therefore $  \diag(G/H)\times \{(gh,g)\:|\: h\in L_{0},g\in G\}$ is a coarse entourage   
 of $(G/H)_{min,min}\otimes G_{can,max}$. We then use that    
  $$(\id_{X},i\circ p)(\diag(L\cup X)) \subseteq \diag(G/H)\times \{(gh,g)\:|\: h\in L_{0}, g\in G\}\, .$$  
  
 If $E$ is any equivariant coarse homology theory, 
 then  the canonical morphism $$E(X)\to \colim_{L\in \mathrm{LF}((G/H)_{min,min}\otimes G_{can,max}) } E(L)$$
  is an equivalence since the elements of $ \mathrm{LF}((G/H)_{min,min}\otimes G_{can,max}) $ containing $X$ are cofinal and for {those} elements the 
 inclusions $X\to   L$ are coarse equivalences. Since we assume in addition that 
  $E$  is continuous, the canonical morphism      $$  \colim_{L\in \mathrm{LF}((G/H)_{min,min}\otimes G_{can,max}) } E(L)\to E((G/H)_{min,min}\otimes G_{can,max})$$ is an equivalence. Hence  the composition of these equivalences is an equivalence
\[E(X)\to E((G/H)_{min,min}\otimes G_{can,max})\, . \qedhere\]
\end{proof}
 
   
  
Using the inclusion
\begin{equation}\label{evrwvjoivjiowewrervewrv}
i \colon X\times G \to  G/H\times G\to Z_{0}
\end{equation}
(see \eqref{fqewqewfkqwpefowqefewfef} for the notation $Z_{0}$ {as a subspace of ${\mathcal{O}}_{\tau}((G/H)_{min,min})\otimes G_{can,max}$}  and \eqref{rgfqfewfqewf} for $\bC_{\tau}(G/H)$) we get an  inclusion 
\begin{equation}\label{evrwvjoivjiowewrervewrv234}
i_{*} \colon  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)\to \bC_{\tau}({(G/H)_{min,min}})
\end{equation}
which identifies $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$
  with the full subcategory of objects of $ \bC_{\tau}({(G/H)_{min,min}}) $ supported on  $i(X)$. 
   
   
    In the following $
    \mathrm{Idem}(\mathrm{Res}^{G}_{H}(   {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H ) $
 is
 the relative idempotent completion using the embedding of $\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H$ as an ideal into $\mathrm{Res}^{G}_{H}(  \bC^{(G) }_{\mathrm{std}})\rtimes H $, \cite[Sec.~15]{cank}.)
Recall the notation for morphisms in crossed {products} from   \cite[Def.~5.1]{crosscat}. 
    \begin{ddd}\label{weoigjowergrgweerffrfwff}
We define the functor 
   \begin{equation}\label{qwefewfefefweqfewfqwef}
\Theta \colon  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)\to  \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H ) 
\end{equation}
   as follows:
 \begin{enumerate}
 \item objects: $\Theta$ sends the object $(C,\rho,\mu)$ in $  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ to the object
 $( C,\rho ,{\pi} )$ in the category $\mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H)$,
 where the orthogonal projection  $\pi$ on $(C,\rho)$ is given by  \begin{equation}\label{erergwgerw4r43r3}
\pi: =\frac{1}{|H|}\sum_{h\in H} (\mu(H),h)\, .
\end{equation}
  \item \label{qewrglijqeovewfwerfwerfwerferfw} morphisms:    $\Theta$ sends $A \colon (C,\rho,\mu)\to (C',\rho',\mu')$ in $   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ to the morphism $$ (A,e) \colon ( C,\rho,  \pi)
\to ( C',{\rho'} ,\pi')$$
in $\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H$.   
\end{enumerate}
\end{ddd}
In order to see that $\Theta$ is well-defined on morphisms   
     note that the relations    $A\mu(H)=\mu'(H)A$ and   $h\cdot A=A$ for all $h$ in $H$  imply 
  \begin{equation*}
(A,e)\pi=\pi'(A,e)=\frac{1}{|H|}\sum_{h\in H}(\mu'(H)A,h)\, .
\end{equation*}
  The first  equality implies that $\Theta$ is compatible with the composition and the involution.
  
    
 \begin{prop}\label{wqeroighowergergregrgwergwreg}
 The morphism $p^{\bC}_{G/H}$ in \eqref{verwejvweopvwerverwv} is an equivalence if and only if the morphism
  $K^{C^{*}\mathrm{Cat}}(\Theta) $ is  an equivalence, where $\Theta$ is as  in Definition \ref{weoigjowergrgweerffrfwff}. \end{prop}
  \begin{proof}
   In analogy to the diagonal functor \eqref{rgergergegwegerg} we define 
  \begin{eqnarray}\label{rgergergegffewefwefwefewfwfwegerg}
\delta' \colon \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))
&\stackrel{C_{0}(G/H)\otimes -}{\to}&
 \mathrm{KK}^{G}(C_{0}(G/H) ,C_{0}(G/H) \otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)) \, .\nonumber
\end{eqnarray}
We then have a commuting diagram
\begin{align}
\label{erjioqergqrfqwefwqefqwfqef}\\
\mathclap{
\xymatrix{
\mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar@/^-4cm/[dd]_{p'}\ar[rr]^{\mathrm{KK}(\C,i_{*})}\ar[d]^{\delta'} && \ar@/^4cm/[dd]^{p^{\bC}_{G/H}}  KK(\C,\bC_{\tau}(G/H))\ar[d]^{\delta_{G/H}^{\bC}}\\
\mathrm{KK}^{G}(C_{0}(G/H) ,C_{0}(G/H) \otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rr]^{C_{0}(G/H)\otimes i_{*}}
\ar[d]^{\mu'} &&           \mathrm{KK}^{G}(C_{0}(G/H) ,C_{0}(G/H) \otimes \bC_{\tau}(G/H) )\ar[d]^{\mu^{\bC}_{G/H}}\\
\mathrm{KK}^{G}(C_{0}(G/H), {\mathbf{K}}^{(G)}_{\mathrm{std}})\ar@{=}[rr] && \mathrm{KK}^{G}(C_{0}(G/K),   {\mathbf{K}}^{(G)}_{\mathrm{std}})
}
}
\notag
\end{align}
where $\mu' \coloneqq \mu^{\bC}_{G/H}\circ ( C_{0}(G/H)\otimes i_{*})$ {and $i_*$ is as in \eqref{evrwvjoivjiowewrervewrv234}.}
The filler of the upper square is induced from the fact that \eqref{saCWDQKOIOJ1} is a bifunctor. Implicitly we also   used the Lemma \ref{glkbijowerfvvfevsdfvsfdv} in order to relate $\hatotimes$ and $\otimes$.


\begin{lem}\label{wegiojergwergwrefrfref}
The morphism $\mathrm{KK}(\C,i_{*}) \colon \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\to  \mathrm{KK}(\C,\bC_{\tau}(G/H))$ is an equivalence.
\end{lem}
\begin{proof}
 Using the definitions
$K\bC\cX^{G}_{c}(-):=K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(-))$ and $K^{C^{*}\mathrm{Cat}}(-):=\mathrm{KK}(\C, -)$  and \eqref{evrwvjoivjiowewrervewrv} we can rewrite the morphism in question as 
$$K\bC\cX^{G}_{c}(X)\to K\bC\cX^{G}_{c}((G/H)_{min,min}\otimes G_{can,max})\to K^{C^{*}\mathrm{Cat}} (\bC_{\tau}(G/H))$$
where the morphisms are induced by the canonical inclusion of $C^{*}$-categories.
We have seen in the proof of Proposition \ref{wrtoihgjoergergegwgrgwerg}  that the second morphism (the left vertical morphism in \eqref{qfwefeeqfeqfqwfqewwfeqwf}) is an equivalence. The first morphism is induced by the inclusion $X\to (G/H)_{min,min}\otimes G_{can,max}$. Since $ K\bC\cX^{G}_{c}$ is a continuous equivariant coarse homology theory this follows from Lemma \ref{wegoijwegowiergwegerw}.
  \end{proof}
{We {continue with} the proof of Proposition \ref{wqeroighowergergregrgwergwreg}.} We define
$p':=\mu'\circ \delta'$.
In view of \eqref{erjioqergqrfqwefwqefqwfqef} and Lemma \ref{wegiojergwergwrefrfref} 
we conclude that
\begin{equation}\label{eq14toirj13o}
p'\simeq p^{\bC}_{G/H}\, .
\end{equation}



 We consider the morphism $\epsilon \colon \C\to \C\rtimes H$ which sends $1$ to the projection $\frac{1}{|H|}\sum_{h\in H} (1,h)$
 see \cite[(4.21)]{KKG}.  
We have the following commuting diagram.
\begin{align}
\label{oigjeroigergergergergwe}\\
\mathclap{
\xymatrix{\mathrm{KK}^{G}(C_{0}(G/H),C_{0}(G/H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[r]^-{\mu'}\ar[d]\ar@/_5.3cm/[dd]_-{r^{G}_{H}}^{\simeq}& \mathrm{KK}^{G}(C_{0}(G/H),  {\mathbf{K}}^{(G)}_{\mathrm{std}}) \ar[d]\ar@/^4cm/[dd]_{r^{G}_{H}}^{\simeq}\\ 
\mathrm{KK}^{H}(\mathrm{Res}^{G}_{H}(C_{0}(G/H)), \mathrm{Res}^{G}_{H}(C_{0}(G/H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)))\ar[r]^-{\mathrm{Res}^{G}_{H}(\mu')}\ar[d]^{\iota^{*}}&\mathrm{KK}^{H}(\mathrm{Res}^{G}_{H}(C_{0}(G/H)),\mathrm{Res}^{G}_{H}(   {\mathbf{K}}^{(G)}_{\mathrm{std}}))\ar[d]^{\iota^{*}}\\
\mathrm{KK}^{H}(\C, \mathrm{Res}^{G}_{H}(C_{0}(G/H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)))\ar@/_5.3cm/[dd]_{j^{H}}^{\simeq}\ar[r]^-{\mathrm{Res}^{G}_{H}(\mu')}\ar[d]^{-\rtimes H}&\ar@/^4cm/[dd]_{j^{H}}^{\simeq}\mathrm{KK}^{H}(\C,\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G)}_{\mathrm{std}}))\ar[d]^{-\rtimes H }\\\mathrm{KK}(\C\rtimes H, (\mathrm{Res}^{G}_{H}(C_{0}(G/H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)))\rtimes H)\ar[r]^-{\mathrm{Res}^{G}_{H}(\mu')\rtimes H}\ar[d]^{\epsilon^{*}}&\mathrm{KK}(\C\rtimes H,\mathrm{Res}^{G}_{H} (    {\mathbf{K}}^{(G) }_{\mathrm{std}}) \rtimes H)\ar[d]^{\epsilon^{*}}\\\mathrm{KK}(\C, (\mathrm{Res}^{G}_{H}(C_{0}(G/H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)))\rtimes H)\ar[r]^-{\mathrm{Res}^{G}_{H}(\mu')\rtimes H} &\mathrm{KK}(\C,\mathrm{Res}^{G}_{H}(     {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H)
}
}
\notag
\end{align}
Here $\iota \colon \C\to \mathrm{Res}^{G}_{H}(C_{0}(G/H))$ is the homomorphism sending $z$ in $\C$ to $z\chi_{H}$, where $\chi_{H}$ is the characteristic function of the orbit $H$ in $G/H$ (this is the value  of \cite[(4.3)]{KKG} at $B=\C$). 
The second and the last middle {square} commute by the associativity of the composition in $\mathrm{KK}^{H}$ and $\mathrm{KK}$, respectively. The first and the third {square} commute since
$\mathrm{Res}^{G}_{H}
$ and $-\rtimes H$ are functors.
For the fact that $r^{G}_{H}$ and $j^{H}$ are equivalences we refer to the spectrum-valued version of
the Green--Julg Theorem \cite[Thm.~1.22]{KKG}. 


We furthermore have the diagram

\begin{align}
\label{efqefeqfqefefqewf}\\
{\scriptsize\mathclap{
\xymatrix{ \Hom_{\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} )}(C_{0}(G/H),C_{0}(G/H))\times \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[r]^-{{\hat  \otimes}}\ar[d]&\mathrm{KK}^{G}(C_{0}(G/H),C_{0}(G/H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)) \ar[d] \ar@/^4.3cm/[dd]_{r^{G}_{H}}^{\simeq}\\ 
\Hom_{\Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )}(\mathrm{Res}^{G}_{H}C_{0}(G/H),\mathrm{Res}^{G}_{H}C_{0}(G/H))\times \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[r]^-{ {\hat  \otimes}}\ar[d]^{\iota^{*}\times \id} &\mathrm{KK}^{H}(\mathrm{Res}^{G}_{H}(C_{0}(G/H)), \mathrm{Res}^{G}_{H}(C_{0}(G/H))\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[d]^{\iota^{*}} \\
\Hom_{\Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )}(\C,\mathrm{Res}^{G}_{H}C_{0}(G/H))\times \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[r]^-{{\hat  \otimes}}\ar[d]^{(-\rtimes H)\times \id} &\mathrm{KK}^{H}(\C, \mathrm{Res}^{G}_{H}(C_{0}(G/H))\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)) \ar[d]^{-\rtimes H}\ar@/^4.3cm/[dd]_{j^{H}}^{\simeq} \\
\Hom_{C^{*}\mathbf{Alg}^{\mathrm{nu}} }(\C\rtimes H,\mathrm{Res}^{G}_{H}C_{0}(G/H)\rtimes H) \times \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[r]^-{ {\hat  \otimes}}\ar[d]^{\epsilon^{*}\times \id} &\mathrm{KK}(\C\rtimes H, (\mathrm{Res}^{G}_{H}(C_{0}(G/H))\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\rtimes H)  \ar[d]^{\epsilon^{*}} \\
\Hom_{C^{*}\mathbf{Alg}^{\mathrm{nu}} }(\C,\mathrm{Res}^{G}_{H}C_{0}(G/H)\rtimes H) \times \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[r]^-{{\hat  \otimes} }&\mathrm{KK}(\C, (\mathrm{Res}^{G}_{H}(C_{0}(G/H))\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\rtimes H)
}}}\notag
\end{align}
In the targets of the two lower maps  we implicitly used the identification \begin{equation}\label{eqwffiouhq9wefewfqfewf}
(A\rtimes H)\otimes {\mathbf{B}}\cong  (A\otimes {\mathbf{B}}) \rtimes H 
\end{equation}  for
$A$ in $\Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ and ${\mathbf{B}}$ in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$.
The second and the last {square} commute since  ${\hat  \otimes}$  in \eqref{saCWDQKOIOJ1} is a bifunctor. 
We now provide the fillers for the first and the third {square}. 
We consider the diagram 
$$\xymatrix{ \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times  C^{*}\mathbf{Alg}^{\mathrm{nu}}\ar[r]^-{\otimes}\ar[d]^{\mathrm{Res}^{G}_{H}\times \id}&\ar[d] \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}) \ar[r]^-{\mathrm{kk}^{G}}\ar[d]^{\mathrm{Res}^{G}_{H}}&\mathrm{KK}^{G} \ar[d]^{\mathrm{Res}^{G}_{H}} \\  \Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times  C^{*}\mathbf{Alg}^{\mathrm{nu}} \ar[r]^-{\otimes}& \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ar[r]^-{\mathrm{kk}^{H}}& \mathrm{KK}^{H}  }$$ 
The left cell obviously commutes, and the right cell commutes 
by \cite[Thm.\ 1.21]{KKG}.
 We now extend using the universal property of $\mathrm{kk} \colon 
  C^{*}\mathbf{Alg}^{\mathrm{nu}}\to \mathrm{KK}$ \cite[Thm.\ 1.18]{KKG}  in order to get a commuting diagram
  $$\xymatrix{ \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times  \mathrm{KK} \ar[r]^-{  {\hat  \otimes}}\ar[d]^{\mathrm{Res}^{G}_{H}\times \id}&\ar[d] \mathrm{KK}^{G} \ar[d]^{\mathrm{Res}^{G}_{H}}  \\  \Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times  \mathrm{KK}   \ar[r]^-{ {\hat  \otimes}}& \mathrm{KK}^{H}   }$$
  This applied to morphism spaces yields the filler of the first  middle square in 
  \eqref{efqefeqfqefefqewf}. 
  In order to justify the third middle square we argue similarly.
We consider the diagram 
$$\xymatrix{ \Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times  C^{*}\mathbf{Alg}^{\mathrm{nu}}\ar[r]^-{\otimes }\ar[d]^{ -\rtimes H\times \id}&  \Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}}) \ar[r]^-{\mathrm{kk}^{G}}\ar[d]^{ - \rtimes H}&\mathrm{KK}^{H} \ar[d]^{-  \rtimes H} 
\\  \Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times  C^{*}\mathbf{Alg}^{\mathrm{nu}}  \ar[r]^-{\otimes}& \Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\ar[r]^-{\mathrm{kk}}& \mathrm{KK}  }$$ 
The left square commutes  because of \eqref{eqwffiouhq9wefewfqfewf}, and the right cell commutes by \cite[Thm. 1.21]{KKG}. 
We now extend using the universal property of $\mathrm{kk}: 
  C^{*}\mathbf{Alg}^{\mathrm{nu}}\to \mathrm{KK}$  in  \cite[Thm.\ 1.18]{KKG} in order to get a commuting diagram 
$$\xymatrix{ \Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times\mathrm{KK} \ar[r]^-{ {\hat  \otimes} }\ar[d]^{ -  \rtimes H\times \id}& \mathrm{KK} \ar[d]^{ -  \rtimes H} 
\\  \Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}} )\times \mathrm{KK}  \ar[r]^-{ {\hat  \otimes}}&\mathrm{KK}  }$$
This square yields the of the  third middle square in
  \eqref{efqefeqfqefefqewf}. 
  
  We specialize the diagram \eqref{efqefeqfqefefqewf} at $\id_{C_{0}(G/H)}$ in
  $ \Hom_{\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} )}(C_{0}(G/H),C_{0}(G/H))$. Then we get 
\begin{align}
\label{efqefeqfqfvdfvdfvefefqfwefwefewf}
\xymatrix{  \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rr]^-{ \delta'}\ar@{=}[d]&&\mathrm{KK}^{G}(C_{0}(G/H),C_{0}(G/H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)) \ar[d] \ar@/^5.35cm/[dd]_{r^{G}_{H}}^{\simeq}\\ 
  \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rr]^-{\id_{\mathrm{Res}^{G}_{H}C_{0}(G/H)}{\hat  \otimes}}\ar@{=}[d]  &&\mathrm{KK}^{H}(\mathrm{Res}^{G}_{H}(C_{0}(G/H)), \mathrm{Res}^{G}_{H}(C_{0}(G/H))\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[d]^{\iota^{*}} \\
  \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rr]^-{\iota {\hat  \otimes}}\ar@{=}[d] &&\mathrm{KK}^{H}(\C, \mathrm{Res}^{G}_{H}(C_{0}(G/H))  \otimes   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)) \ar[d]^{-\rtimes H}\ar@/^5.35cm/[dd]_{j^{H}}^{\simeq} \\
 \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rr]^-{( \iota\rtimes H) {\hat  \otimes}}\ar@{=}[d]  &&\mathrm{KK}(C^{*}(H), (\mathrm{Res}^{G}_{H}(C_{0}(G/H))  \otimes   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\rtimes H)  \ar[d]^{\epsilon^{*}} \\
  \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rr]^-{  \delta''}&&\mathrm{KK}(\C, (\mathrm{Res}^{G}_{H}(C_{0}(G/H)  \otimes   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\rtimes H)  }
\end{align}
where  
\begin{align*}
\delta'' \coloneqq {\epsilon^{*}}(\iota\rtimes H){\hat  \otimes}- \colon \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)) & \to \mathrm{KK}(\C, (\mathrm{Res}^{G}_{H}(C_{0}(G/H)) \rtimes H)\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)))\\
&\simeq  \mathrm{KK}(\C, (\mathrm{Res}^{G}_{H}(C_{0}(G/H))\otimes  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\rtimes H)    \, .
\end{align*}
Composing \eqref{efqefeqfqfvdfvdfvefefqfwefwefewf} with \eqref{oigjeroigergergergergwe} we get a commuting square
$$\xymatrix{  \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rrr]^-{p'=  \mu' \circ \delta'}\ar@{=}[d]&&&  \mathrm{KK}^{G}(C_{0}(G/H),   {\mathbf{K}}^{(G)}_{\mathrm{std}})\ar[d]_{\simeq}^{j^{H}\circ r^{G}_{H}}\\  \mathrm{KK}(\C, \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X))\ar[rrr]^-{p'':=\mathrm{Res}^{G}_{H}(\mu'\rtimes H)\circ \delta''}
&&&\mathrm{KK}(\C,\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H)
}$$
 We therefore have an equivalence   \begin{equation}\label{qwfewpeofkqwepfqew1f}
p''\simeq p'\stackrel{\eqref{eq14toirj13o}}{\simeq} p^{\bC}_{G/H}\, .
\end{equation}
 By construction the morphism $p''$ is induced 
 by an explicit  functor 
 \begin{equation}\label{vervevervwekvpewvervevwev}
\Theta' \colon  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)\to \mathrm{Res}^{G}_{H}( {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H\, .
\end{equation}
 Inserting all definitions we see that $\Theta'$  is  given by follows:
 \begin{enumerate}
 \item objects:  $\Theta'$ sends the object $(C,\rho,\mu)$ in $  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ to
 $(C, \rho  )$ in $ \mathrm{Res}^{G}_{H}( \tilde {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H$.
\item morphisms:  The functor $\Theta'$ sends a morphism $A \colon (C,\rho,\mu)\to (C',\rho',\mu')$ in $   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ to the morphism $$\frac{1}{|H|}\sum_{h\in H}(\mu'(H)A,h) \colon ( C,\rho  )
\to  (C',\rho' )$$
in $\mathrm{Res}^{G}_{H}( {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H$.
Note that {$\mu'(H)A=A\mu(H)$}.  
 \end{enumerate}
 Note that $\Theta'$ is not full.
 
 

 Let $$ c:\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H\to \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H )$$ be the inclusion.
We consider the two  functors  $$ \Theta , c\circ \Theta':  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)\to   \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G) }_{\mathrm{std}})\rtimes H))$$
 in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$.
 
\begin{lem}\label{egeroigjregwergege}
 We have an equivalence
 \begin{equation}\label{werfpokfpwerferffwrf}
K^{C^{*}\mathrm{Cat}} ( \Theta )\simeq K^{C^{*}\mathrm{Cat}} ( c\circ \Theta' ) \colon K^{C^{*}\mathrm{Cat}} (  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X ))\to K^{C^{*}\mathrm{Cat}}(\mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H)) \, .
\end{equation} 
\end{lem}
\begin{proof}
We construct a {relative Murray von Neumann equivalence (see  \cite[Def.\ 15.9]{cank} for this notion)}  $v:\Theta\to c\circ \Theta'$.
It is given by the family $(v_{(C,\rho,\mu})_{(C,\rho,\mu)\in   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)}$ of partial isometries, where 
$v_{(C,\rho,\pi)} \colon (C,\rho,p)\to (C,\rho)$ is the canonical inclusion. 

Since the formulas for the actions of $\Theta$ and $\Theta'$ are equal this family is indeed a natural transformation. We now apply \cite[Prop.\ 15.10]{cank} which gives the equivalence \eqref{werfpokfpwerferffwrf}.
\end{proof}

{We  {continue with} the proof of Proposition \ref{wqeroighowergergregrgwergwreg}.} We now note that $$K^{C^{*}\mathrm{Cat}}(c)\colon  K^{C^{*}\mathrm{Cat}} (\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H)\to K^{C^{*}\mathrm{Cat}}(\mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H))$$
is an equivalence {by the Morita invariance of $K^{C^{*}\mathrm{Cat}}$ \cite[Thm.\ 14.14]{cank}}. 

 Therefore
$K^{C^{*}\mathrm{Cat}}(\Theta) $ is an  equivalence if and only if $ K^{C^{*}\mathrm{Cat}}(\Theta' )$ is an  equivalence. 
The Proposition \ref{wqeroighowergergregrgwergwreg} now follows from 
the
   combination of  \eqref{qwfewpeofkqwepfqew1f} and the fact that $p''$ is induced by the functor $\Theta'$.
  \end{proof}
  
 Recall the Definition \ref{weoigjowergrgweerffrfwff} of the functor $\Theta$ and that $H$ denotes a finite subgroup of $G$.
  
  The next proposition finishes the proof of 
 Proposition  \ref{qeoigjqergqwedewdqed} and hence of Theorem \ref{qreoigjoergegqrgqerqfewf}.
  Recall Definition  \ref{wefiqrjhfoifewqdqwedqwedqewd} of the condition on ${\mathbf{K}}$ of being compatible with orthgonal sums.
  \begin{prop}\label{4oijotrherhrthrhe}
   If ${\mathbf{K}}$ is compatible with orthogonal sums, then 
  $K^{C^{*}\mathrm{Cat}}(\Theta)$ is an equivalence.
  \end{prop}

The proof of this Proposition \ref{4oijotrherhrthrhe} is based on the factorization of $\Theta$ as described by the commutative diagram \eqref{oijiojiovawvasfvvsdvdsva}. The functors in this diagram will all induce equivalences in $K$-theory, but for different reasons. {The rest of this section is devoted to the proof of Proposition \ref{4oijotrherhrthrhe} {which} is split in several lemmas.}


   \begin{lem}\label{qwroiqrgegwregrgwregwg}
 The functor $\Theta$ is fully faithful.
 \end{lem}
 \begin{proof}
 Recall that   $X=G(H,e)$ is a  subspace  of $(G/H)_{min,min}\otimes G_{can,max}$, see  \eqref{qrewfpokfpowefqweffqwfwefq}.  Let $(C,\rho,\mu)$ and $ (C',\rho',\mu')$ be objects of $  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$.
 Then $\Theta(C,\rho,\mu)=(C,\rho,\pi)$  with $\pi$ given by \eqref{erergwgerw4r43r3}, and similarly $\Theta(C',\rho',\mu')= (C',\rho',\pi')$.
 Let $$B:(C,\rho,\pi)\to (C',\rho',\pi')$$
 by any morphism. We can write $B=\sum_{h\in H} (B_{h},h)$, where $B_{h}:C\to C'$. The condition
 $\pi'B\pi=B$ implies that
 $B_{h}= \mu'(H)B_{e}\mu(H)$ and $h \cdot B_{e}=B_{e}$ for every $h$ in $H$.  We can define the  morphism 
\begin{equation}\label{qewfqowifheiofewfqewfewf}
A:=\frac{1}{|H|}\sum_{g \in G}  g\cdot B_{e}:(C,\rho,\mu)\to (C',\rho',\mu')
\end{equation}  
in      $  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$.
 Then $\Theta(A)=B$.
 The formula \eqref{qewfqowifheiofewfqewfewf} defines an inverse of $\Theta$ on the level of morphisms.
  \end{proof}


 
 
 In $ \mathrm{Idem}(\mathrm{Res}^{G}_{H}(   {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H)$
 we consider the full subcategory $\bD$  of objects of the form 
 $(C,\rho,(\mu(H),e))$, where $(C,\rho,\mu)$ is in ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(X)$, see Definition \ref{qrohwoifewfqwefqwfewqf}.
 We let furthermore
 $\bD'$ be the subcategory {of  $ \mathrm{Idem}(\mathrm{Res}^{G}_{H}( {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H)$} on objects of the form $(C,\rho,(\mu(Z),e))$,
 where $(C,\rho,\mu)$ is in $ {\mathbf{C}}^{(G)}_{\mathrm{lf}}(Y)$ for some free $G$-set $Y$ and $H$-invariant subset $Z$ of $Y$.  {By $\Lambda$ we denote the
 canonical inclusion of $\bD$ into $\bD'$. The idempotent completion  of  $\bD'$ is  formed relative to $ \mathrm{Idem}(\mathrm{Res}^{G}_{H}(   \bC^{(G)}_{\mathrm{std}})\rtimes H)$.}
 Then we have the following diagram \begin{equation}\label{oijiojiovawvasfvvsdvdsva}
\xymatrix{\ar@/^1cm/@{..>}[rrrr]^{\Theta}  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X) \ar[r]^{\Phi}&\mathrm{Idem}(\bD)\ar@{..>}[rr]^{\mathrm{Idem}(\Lambda)} && \mathrm{Idem}(\bD')\ar[r]^-{\Delta} & \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H)\\
&\bD\ar[u]^{\Xi}\ar[rr]^{\Lambda} && \bD'\ar[u]^{\Psi}&}\, ,
\end{equation}
where $\Delta$ is again  the canonical inclusion.
The upper line is then a factorization of $\Theta$ as indicated.
 
 
 In the following we will show that  all bold  morphisms in \eqref{oijiojiovawvasfvvsdvdsva} induce equivalences after applying $K^{C^{*}\mathrm{Cat}}$.
  It is clear that this implies  that   $K^{C^{*}\mathrm{Cat}}(\Theta) $ is  an equivalence.  
 To this end
we   use that $K^{C^{*}\mathrm{Cat}}$ sends Morita equivalences, relative Morita equivalence, relative equivalences, idempotent completions of ideals,  and weak Morita equivalences (see \cite[Sec.\ 14.--16]{cank}) to equivalences. 
In the following lemmas we argue case by case that all bold arrows have one of these properties. 

\begin{lem}\label{wotihgjogregwgregw}
 $\Xi$   is a Morita equivalence.
\end{lem}
\begin{proof}
The $C^{*}$-category category $\bD$ is  unital since $H$ is finite and hence the projections $\mu(H)$ appearing in the description of the objects $(C,\rho,(\mu(H),e))$ of $\bD$ belongs to ${\mathbf{K}}$. 
 Since $\Xi$ is the canonical map into the idempotent completion it is then a Morita equivalence.
\end{proof}

 

\begin{lem}\label{qergpoqjekrgpgqgre}
 $\Delta$  
is a relative equivalence.
 \end{lem}
\begin{proof} By definition of $\mathrm{Idem}(\bD')$ as a relative idempotent completion
we have a pull-back diagram \begin{equation}\label{qwefoijqiowejfoqwefqefef}
\xymatrix{ \mathrm{Idem}(\bD')\ar[r]\ar[d] &\bE  \ar[d]^{\Delta}  \\
  \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  {\mathbf{K}}^{(G)}_{\mathrm{std}})\rtimes H)\ar[r]& \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  \bC^{(G)}_{\mathrm{std}})\rtimes H)  }
\end{equation}
where $\bE$ is the full subcategory of $ \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  \bC^{(G)}_{\mathrm{std}}){\rtimes H})$  
with the same objects as $\mathrm{Idem}( \bD')$.
By construction, the right vertical arrow  {in \eqref{qwefoijqiowejfoqwefqefef}} is fully faithful.
 Furthermore, since
$\bE$ contains the objects of the form 
 $(C,\rho,(\mu(Y),e))$ for all   free $G$-sets $Y$ and all $(C,\rho,\mu)$  in $ {\mathbf{C}}^{(G)}_{\mathrm{lf}}(Y)$, it also contains all  orthogonal {sums}  of these objects. Therefore the right vertical arrow   in \eqref{qwefoijqiowejfoqwefqefef} is essentially surjective and hence is a unitary equivalence. Consequently, the left
vertical morphism in \eqref{qwefoijqiowejfoqwefqefef}   is a relative  equivalence  {(see \cite[Def.\ 15.4]{cank}).}
\end{proof}

\begin{lem}
 $\Psi$ is a relative Morita equivalence.
\end{lem}
\begin{proof}
The category $\bD'$ is an ideal in the   category denoted by $\bE$ in the proof of Lemma \ref{qergpoqjekrgpgqgre} {and is} obtained by considering the same objects, but inheriting the morphisms from  $ \mathrm{Idem}(\mathrm{Res}^{G}_{H}(  \bC^{(G)}_{\mathrm{std}})\rtimes H)$.  
Hence $\Psi$  is an idempotent completion of an ideal in the sense of \cite[Def.\ 15.1]{cank}.
\end{proof}
 

\begin{lem}\label{wetiogwtgwgreregwr}
$\Phi$  is a Morita equivalence.
\end{lem}
\begin{proof}  
The functors $\Lambda$ and $\Delta$ are  are fully faithful by definition. It follows that also $\mathrm{Idem}(\Lambda)$ is fully faithful. Since 
 $ \Theta$ is fully faithful by Lemma \ref{qwroiqrgegwregrgwregwg},  we can conclude  that $\Phi$ is fully faithful, too.

Let $(C,\rho,\mu)$ be an object of $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$. Then we define 
  $$U:=\frac{1}{\sqrt{|H|}} \sum_{h\in H}(\mu(\{h\})  ,h)$$ in $\End_{\bD}((X,(C,\rho,\mu),(\chi(H),e)))$.
We calculate that 
$$UU^{*}= (\mu(\{e\}),e)\, , \quad U^{*}U=\pi\, ,$$
{where $\pi$ is as in \eqref{erergwgerw4r43r3}.}
This calculation shows that the projection 
$\pi$ is {MvN}-equivalent to $(\mu(\{e\}),e)$.
For $h$ in $H$ we consider the unitary $V_{h}:=(\mu(H),h^{-1})$ in $\End_{\bD}(C,\rho,({\mu}(H),e))$.
 Then  $$V_{h}(\mu(\{e\}),e) V_{h}^{*}=(\mu(\{h\}),e)\ .$$
So the projection $(\mu({\{h^{-1}\}}),e)$ is also MvN-equivalent to $\pi$ for every $h$ in $H$.
Since the projections
$((\mu(\{h\}),e))_{h\in H}$  are mutually orthogonal and
$\sum_{h\in H} (\mu(\{h\}),e)=(\mu(H),e)$
we see that any object of $\bD$ is an orthogonal summand  of a finite orthogonal  sum of objects in the essential image of $\Phi$.  This implies that also every object of $\mathrm{Idem}(\bD)$ is an orthogonal summand of a   finite orthogonal  sum of objects in the essential image of $\Phi$. 
\end{proof}


For the next lemma we need the additional assumption on ${\mathbf{K}}$.


\begin{lem}\label{ergoijreogwergegwgwrgwr} If ${\mathbf{K}}$ is compatible with orthogonal sums, then 
  $\Lambda$ is a weak Morita equivalence.
\end{lem}
\begin{proof}
{The functor $\Lambda$  fully faithful is by definition. In the proof of Lemma \ref{wotihgjogregwgregw} we have seen that $\bD$ is unital.
It remains to  show} that the set of objects of $\bD$ is weakly generating in $\bD'$, see \cite[Def.\ 16.1]{cank}.


Let $(C,\rho,(\mu(Z),e))$ be any object of $\bD'$, where $(C,\rho,\mu)$ is in $   {\mathbf{C}}^{(G)}_{\mathrm{lf}}(Y)$ for some free $G$-set $Y$ and $Z$ is a $H$-invariant subset of $Y$.
 Let $y$ be a point in $Y$.
Then we can form the object $(C,\rho,(\mu(Hy),e))$ in $\bD'$.
 We claim that this object is isomorphic to an object in $\bD$.
 We consider the  $G$-equivariant   injection $i:X\to Y$ which sends $(H,e)$ to $y$. 
 We choose an image $u:C'\to C$ of the projection $\mu(Gy)$. 
 Then we define 
 $(C',\rho',\mu')$ in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X)$ such that
 $\rho'_{h}=u^{*}\rho_{h}u$ for every $h$ in $H$ and $\mu'(W)=u^{*}\mu(i(W)) u$ for every subset $W$ of $X$.
  Then we have an isomorphism
 $$(u,e) \colon (C',\rho',(\mu'(H),e))\to (C,\rho,(\mu(Hy),e))$$
 in  $\bD'$. 
 More generally, if $Z$ is any finite $H$-invariant subset of $Y$ (note that $H$ is finite), then $(C,\rho,(\mu(Z),e))$ is isomorphic to a finite sum of objects in $\bD$.
 
  
Let now $(A_{j})_{j\in J}$ with $A_{j} \colon (C_{j},\rho_{j},p_{j})\to (C,\rho,p) 
 $ be a finite family of morphisms in $\bD'$ 
 
 Let  $\epsilon$ in $(0,\infty)$ be given.
 Since ${\mathbf{K}}$ is compatible with orthogonal sums and
 $C$ is  isomorphic to the orthogonal sum of the family of 
 projections $(\mu(S))_{S\in H\backslash Y}$ there exists a finite $H$-invariant subset $Z$ of
  $Y$ such that $$\|A_{j}- (\mu(Z),e)A_{j}\|<\epsilon$$ for all $j$ in $J$.
 \end{proof}
This finishes the proof of Proposition \ref{4oijotrherhrthrhe} and therefore the proofs of the Theorems \ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf} and  \ref{qreoigjoergegqrgqerqfewf}.\ref{wrtigjiogowrefwerfewrf} are now also complete.\phantomsection\label{wefgiojwoegergregegw}
 

\section{Discussion and examples}\label{weogiwjiergreggwrgwerg}


\newcommand{\mathrm{pcc}}{\mathrm{pcc}}
 

The domain of the Paschke morphism is given by  the functor 
 $$K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(-)) \colon G\mathbf{UBC}\to \Sp\, .$$
 Our first goal is to describe its values  one nice spaces in terms
 of the equivariant  homology theory $$K\bC^{G} \colon G\mathbf{Orb}\to \Sp$$ introduced in    \eqref{asdvijqoirgfvfsdvsva}, see Definition \ref{qroeigjoqergerqfewewfqewfeqwf} below for the technical description.  Before we state the result in Proposition \ref{weroigjwerogregrgwffe} we first  explain Elemendorf's theorem and how one can evaluate equivariant  homology theories   on $G$-topological spaces.
  
Let $\mathbf{M}$ be a cocomplete stable $\infty$-category.  
In the present paper we adopt the following simple definition which in some sense reverses the history of this notion. \begin{ddd}\label{weigjorgdg}
An equivariant $\mathbf{M}$-valued homology theory  is  a functor 
 $$E \colon G\mathbf{Orb}\to \mathbf{M}\, .$$ 
 \end{ddd}


 Recall that a weak equivalence between topological spaces is a  continuous map which induces a bijection between the sets of connected components and  isomorphisms between the higher homotopy groups at all base points. We have  a functor
 \begin{equation}\label{adsfasdfasdfadsf}
\ell \colon \Top\to \mathbf{Spc}
\end{equation}
   which presents $\mathbf{Spc}$ as the localization of $\Top$ at the weak equivalences. 
We now consider the functor \begin{equation}\label{wefqwefweqdxasdc}
Y^{G}:G\Top\to {\mathbf{PSh}}(G\mathbf{Orb})
\end{equation}
 which sends  
$X$ in $ G\Top$ to the presheaf $$S\mapsto \ell(\Map_{ G\Top}(S_{disc},X))\, ,$$
where $\Map_{ G\Top}(S_{disc},X)$  in $\Top$ is the topological mapping space of equivariant maps. 
A map between $G$-topological spaces is a weak equivalence if it induces weak equivalences between the fixed point subspaces for all subgroups of $G$. 
 \begin{theorem}[Elmendorf's theorem] \label{weiogjodfbsefsffgsfgdfg}The functor $Y^{G}$ presents $ {\mathbf{PSh}}(G\mathbf{Orb})$ as the localization  of $ G\Top $ at the  equivariant weak equivalences. \end{theorem}
   

    By the universal property of  presheaves, the pull-back along the  Yoneda embedding   $\mathrm{yo}:G\mathbf{Orb}\to {\mathbf{PSh}}(G\mathbf{Orb})$ induces  an equivalence 
  $$\mathrm{yo}^{*}:\Fun^{\colim}({\mathbf{PSh}}(G\mathbf{Orb}),\mathbf{M})\stackrel{\simeq}{\to} \Fun(G\mathbf{Orb},\mathbf{M})\ .$$
  Let $E:G\mathbf{Orb}\to \mathbf{M}$ be an equivariant homology theory. Its
 colimit preserving extension  
  to presheaves     is the  left Kan-extension  $\mathrm{yo}_{!}E:{\mathbf{PSh}}(G\mathbf{Orb})\to \mathbf{M}$  of $E$ along $\mathrm{yo}$.
    \begin{ddd} \label{weiogwegerffs}The evaluation of $E$ on $G$-topological spaces is  defined as composition
 (which we will again denote by $E$) \begin{equation}\label{adfiuhuihavfvasdcacsadcasca}
E \colon G\Top  \stackrel{Y^{G}}{\to} {\mathbf{PSh}}(G\mathbf{Orb})\stackrel{\mathrm{yo}_{!}E}{\to} \mathbf{M}\, .
\end{equation}\end{ddd}
 If  $S$ is in $G\mathbf{Orb}$, then the value
 of the original functor  $E$ on $S$ and the evaluation of $E$ on the discrete space  $G$-space $S_{disc}$ coincide so that there is no conflict of notation.
The value of the   equivariant homology theory  on a general  space $X$ in $   G\Top$ is given by the coend
\begin{equation}\label{qwefpojfopwefqwefqwefqew}
E(X):=\int^{G\mathbf{Orb}}  E \wedge  \Sigma^{\infty}_{+} Y^{G}(X)\ ,
\end{equation}
where  
 $\wedge :\mathbf{M}\times \Sp\to \mathbf{M}$ is the tensor structure of $\mathbf{M}$ (the same as \eqref{sdvjqr0evj0fvsfvvfv})  which exists by the cocompleteness and stability assumptions on $\mathbf{M}$.


 We let $G\mathbf{UBC}^{\mathrm{pcc}}$ be the full subcategory of 
$G\mathbf{UBC}$ of $G$-uniform bornological coarse spaces which have the following properties:
\begin{enumerate} 
\item the underlying topological space is Hausdorff
\item  the bornology is generated by relatively compact subsets
\item the coarse structure is generated by all entourages of the form
$G(K\times K)$, where $K$ is a relatively compact connected subset 
\item $G$ acts properly and cocompactly.
\end{enumerate}
The category $G\mathbf{UBC}^{\mathrm{pcc}}$ contains all $G$-finite $G$-simplicial complexes with finite stabilizers with the structures induced by the spherical path metric.
   We consider the functor  
$ \iota: G\mathbf{UBC}\to G\Top$ which takes the underlying $G$-topological space. 
\begin{lem}\label{hfegwiugheifofdsfd} The restriction
$\iota_{|G\mathbf{UBC}^{\mathrm{pcc}}}:   G\mathbf{UBC}^{\mathrm{pcc}} \to G\Top$ is fully faithful.
 \end{lem}
  \begin{proof}
  It is clear that $\iota_{|G\mathbf{UBC}^{\mathrm{pcc}}}$ is faithful. We must show that it is full.
  Let $X,Y$ be in $ G\mathbf{UBC}^{\mathrm{pcc}}$ and $f \colon X\to Y$ be an equivariant continuous map.
  We must show that it is controlled, uniformly continuous and proper.
  
  We first show by contradiction that $f$ is proper.
  Let $K$ be a  relatively compact subset of $Y$ and assume that $f^{-1}(K)$ is not relatively compact.
  Then there exists a net $(x_{\alpha})_{\alpha}$ in $f^{-1}(K)$ without accumulation point in $X$.
 Since $K$ is relatively compact, and $G\backslash X$ is compact,  we can assume by taking a subnet that
 $(f(x_{\alpha}))_{\alpha}$ and  $([x_{\alpha}])_{\alpha}$ converge in $Y$ and $G\backslash X$, respectively.
 By the second condition  there exists a family $(g_{\alpha})_{\alpha}$ in $G$ such that $(g_{\alpha}x_{\alpha})_{\alpha}$ converges. 
 Since then $(g_{\alpha}f(x_{\alpha}))_{\alpha}$ also converges  and $G$ acts properly on $Y$ we can  assume after taking a subnet that
 $(g_{\alpha})_{\alpha}$ is constant. 
 But then   $(x_{\alpha})_{\alpha}$ converges, a contradiction.
  
 We claim that any invariant open entourage of the diagonal of $X$ is uniform. 
 The claim immediately implies that $f$ is uniformy continuous. Indeed, if $V$ is any uniform entourage of $Y$, then there exists an invariant  uniform entourage $V'$ of $Y$ such that $V'\subseteq V$. But then $(f\times f)^{-1}(V')$ is invariant and open, hence a uniform emtourage of $X$. The relation $(f\times f)^{-1}(V')\subseteq (f\times f)^{-1}(V)$ implies that $ (f\times f)^{-1}(V)$ is uniform. 
 
 
 We now show the claim. Assume by contradiction that $U$ is not uniform.
 Then for every invariant uniform entourage $V$ of $X$ there exists 
 $(x_{V},y_{V})$ in $V\setminus U$. By compactness of the quotient we can assume, after taking a cofinal subnet $(V_{\alpha})_{\alpha}$ of uniform entourages,  that 
 $[x_{V_{\alpha}}]\to [x]$ and $[y_{V_{\alpha}}]\to [y]$.  We can find a net $(g_{ \alpha})_{\alpha}$ in $G$ such that $g_{\alpha}x_{V_{\alpha}}\to x$ in $X$.
 But then  also $g_{\alpha}y_{V_{\alpha}}\to x$ since $X$ is Hausdorff and the  net $(V_{\alpha})_{\alpha}$ of uniform entourages is cofinal. Since $U$ is $G$-invariant we have
 $(g_{\alpha}x_{V_{\alpha}},g_{\alpha}y_{V_{\alpha}})\not\in U$ for all $\alpha$, and since $U$ is open we conclude that also $(x,x)\not\in U$.
 But this is impossible since $U$ was an open neighbourhood of the diagonal.  
   
 We check on generators that $f$ is controlled. Let $K$ be a  relatively compact    connected subset of $X$ and consider the generator 
 $G(K\times K)$ of the coarse structure of $X$. Then $f(K)$ is relatively compact and connected, too.
  Therefore
 $(f\times f)G(K\times K) =G(f(K)\times f(K))$ is a coarse entourage of $Y$. 
 \end{proof}

  



   
      
 
Recall that  $K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(-))$ is defined on $G\mathbf{UBC}$. By Lemma \ref{hfegwiugheifofdsfd} 
 we  can restrict $K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(-))$ to a functor defined on the full subcategory $G\mathbf{UBC}^{\mathrm{pcc}}$ of $G\Top$. In contrast, the equivariant homology theory   $K\bC^{G}$ gives rise to a functor defined on all of $G\Top$ by Definition \ref{weiogwegerffs}.
 Therefore, as a preparation we present a  general result which helps to compare a functor with homological  properties defined on some full subcategory of  $G\Top$ with  an associated equivariant homology theory.

Let $\bV$ be a simplicial model category  with weak equivalences  $W$, {homotopy equivalences $W_h$}, and
with functorial  factorizations. 
 The associated $\infty$-category of  $\bV$ is defined by $\bV_{\infty}:=\bV[W^{-1}]$. We let $\ell:\bV\to \bV_{\infty}$ denote the canonical functor.
We furthermore  let $\bV^{\mathrm{cf}}$ denote the full subcategory of cofibrant/fibrant objects in $\bV$. {The following lemma is of course well-known, but for lack of reference, we include a proof here.}

\begin{lem}
The inclusion $\bV^\mathrm{cf} \to \bV$ induces an equivalence of Dwyer--Kan localizations $\bV^\mathrm{cf}[W_h^{-1}] \simeq \bV[W^{-1}]$.
\end{lem}
\begin{proof}
We consider  the following square 
$$\xymatrix{\bV^{\mathrm{cf}}\ar[r]^-{\ell_{h}}\ar[d]&\bV^{\mathrm{cf}}[W^{-1}_{h}]\ar@{..>}[d]\\ \bV\ar[r]^-{\ell}&\bV_{\infty}}$$
where the dotted arrow is obtained from the universal property of the localization $\ell_{h}$. {We claim that it is} an equivalence as desired. In order to produce an inverse 
 we consider the square 
$$\xymatrix{\bV\ar[r]^-{\ell}\ar[d]^{RL}&\bV_{\infty}  \ar@{..>}[d]\\ \bV^{\mathrm{cf}}\ar[r]^-{\ell_{h}}& \bV^{\mathrm{cf}}[W^{-1}_{h}]}$$
where $RL$ is the fibrant-cofibrant replacement functor. 
The dotted arrow is obtained from the universal property
of $\ell$ since $RL$ sends weak equivalences to homotopy equivalences.   
We have a diagram $$\xymatrix{ &L\ar[dr]\ar[dl]& \\ \id &&RL}$$ of endofunctors of $\bV$, where $L$ and $R$ are the fibrant and cofibrant replacement functors. It is 
  sent by $\ell$ to a diagram of equivalences. 
Similarly, $\ell_{h}$ sends  the restriction of this  diagram to $\bV^{\mathrm{cf}}$ to a diagram of equivalences.
From this we can conclude that the two dotted arrows are inverse to each other.
\end{proof}


  


Let $E:\bV\to \mathbf{M}$ be a homotopy invariant functor.


\begin{lem}\label{ewiorgjowergwrewfre}
There exists a functor
$E^{\infty}\colon\bV_{\infty}\to \mathbf{M}$ such that the following square commutes:
$$\xymatrix{\bV^{\mathrm{cf}}\ar[r]^{E_{|\bV^{\mathrm{cf}}}}\ar[d]&\mathbf{M}\\ \bV \ar[r]&\bV_{\infty}\ar[u]_{E^{\infty}}}$$
\end{lem}
\begin{proof}
{We obtain} the desired square from 
$$\xymatrix{\bV^{\mathrm{cf}}\ar[r]^-{\ell_{h}}\ar[d]\ar@/^1cm/[rr]^{E_{|\bV^{\mathrm{cf}}}}&\bV^{\mathrm{cf}}[W^{-1}_{h}]\ar@{..>}[r]\ar[d]^{\simeq}&\mathbf{M}\ar@{=}[d]\\ \bV\ar[r]^{\ell}&\bV_{\infty}\ar[r]^{E^{\infty}}&\mathbf{M}}$$
where the dotted arrow exists since $E$ sends homotopy equivalences to equivalences.
\end{proof}


We consider  some full subcategory  $\bW$      of  $G\Top$  and let 
$E:\bW\to \mathbf{M}$ be some functor. 
We assume that $\cF$ is a family of subgroups of $G$
and that $G_{\cF}\mathbf{Orb}\subseteq \bW$.  
We then define the equivariant homology theory  $E^{\%,\cF} \colon G\mathbf{Orb}\to \mathbf{M}$ as the left Kan extension of the functor  $E_{|G_{\cF}\mathbf{Orb}}$ along $i_{\cF}$:
 $$\xymatrix{G_{\cF}\mathbf{Orb}\ar[rr]^{E_{|G_{\cF}\mathbf{Orb}}}\ar[d]_{i_{\cF}} &&\mathbf{M}\,. \\G\mathbf{Orb} \ar[urr]_{E^{\%,\cF} }&&  }$$
Following Definition \ref{weiogwegerffs} we will consider
$E^{\%,\cF}$ also as a functor $E^{\%,\cF}\colon G\Top\to \mathbf{M}$.

We now use that $G\Top$ admits a simplicial model category structure
with the  weak  equivalences  as described {after \eqref{wefqwefweqdxasdc}} and such that the notion of homotopy is the usual one. By Theorem \ref{weiogjodfbsefsffgsfgdfg} the functor
$Y^{G} \colon G\Top\to {\mathbf{PSh}}(G\mathbf{Orb})$ is equivalent to the functor $G\Top\to G\Top_{\infty}$
 in the notation introduced before Lemma \ref{ewiorgjowergwrewfre}.  {Let $j \colon \bW\to G\Top$ denote the inclusion.}
 \begin{lem} \label{sogipegrggwege}Assume:
\begin{enumerate}
\item $G_{\cF}\mathbf{Orb}\subseteq \bW\subseteq G\Top^{\mathrm{cf}}$
\item  $\bW$ is closed under taking the product with $[0,1]$.
\item  
$E$ is homotopy invariant. 
\end{enumerate}
Then we have a canonical natural transformation of functors
$$j^{*}E^{\%,\cF} \to E \colon \bW\to \mathbf{M}\, .$$
\end{lem}
\begin{proof}
Then we have an equivalence $E\stackrel{\simeq}{\to} j^{*}j_{!}E$ since $j$ is fully faithful.
We claim that $j_{!}E$ is homotopy invariant.  Let 
$X$ be in $G\Top$. Then we must show that $(j_{!}E)([0,1]\times X)\to (j_{!}E)(X)$ is an equivalence.
We use the point-wise formula for the left Kan extension in order to rewrite this as
$$\colim_{(Y\to [0,1]\times X)\in \bW_{/[0,1]\times X}} E(Y)\to \colim_{(Z\to   X)\in \bW_{/  X}}E(Z)\, .$$
We now observe that the maps of the form
$[0,1]\times Z\to [0,1]\times X$ for maps $Z\to X$ are cofinal in the index category of the left colimit. At this point we use that $\bW$ is closed under taking products with an interval.
Indeed, let $(a,b):Y\to [0,1]\times X$ be a map. Then we consider the factorization
$$Y\stackrel{(a,\id_{Y})}{\to}[0,1]\times Y\stackrel{(\id_{[0,1]},b)}{\to} [0,1]\times X\, .$$
Consequently, the morphism above is equivalent to 
$$\colim_{(Z\to X)\in \bW_{/  X}} E([0,1]\times Z)\to \colim_{(Z\to   X)\in \bW_{/  X}}E(Z)\, .$$
This map is an equivalence since $E$ is homotopy invariant. This finishes the proof of the claim.

 By Lemma \ref{ewiorgjowergwrewfre} we get a functor
$(j_{!}E)^{\infty}: {\mathbf{PSh}}(G\mathbf{Orb})\to \mathbf{M}$ fitting into the commutative square in
$$\xymatrix{ &&& \bW\ar[ddl]^(0.65){j} \ar[rrd]^{E}\ar[dl]&& \\G_{\cF}\mathbf{Orb}\ar[d]^{i_{\cF}}\ar[urrr] &&G\Top^{\mathrm{cf}}\ar[d]\ar[rrr]^{(j_{!}E)_{|G\Top^{\mathrm{cf}}}}&&&\mathbf{M}\\ G\mathbf{Orb}\ar[rr]\ar@/_1cm/[rrrrr]_{\mathrm{yo}} &&G\Top \ar[rrr]^{Y^{G}} &&& {\mathbf{PSh}}(G\mathbf{Orb}) \ar[u]^{(j_{!}E)^{\infty}} }$$
Here the   triangle involving $(j_{!}E)_{|G\Top^{\mathrm{cf}}}$  commutes since $j^{*}j_{!}E\simeq E$ as observed already above.
The commutative diagram provides an equivalence $  E_{|G_{\cF}\mathbf{Orb}}   \simeq    i_{\cF}^{*}\mathrm{yo}^{*}(j_{!}E)^{\infty}$.
Applying the left Kan extension $\mathrm{yo}_{!} i_{\cF,!} $ we get an equivalence 
 $$\mathrm{yo}_{!} E^{\%,\cF} \simeq  \mathrm{yo}_{!}  i_{\cF,!} E_{|G_{\cF}\mathbf{Orb}}    \simeq  \mathrm{yo}_{!}   i_{\cF,!} i_{\cF}^{*}\mathrm{yo}^{*}(j_{!}E)^{\infty}   \, .$$
The counit $ \mathrm{yo}_{!}   i_{\cF,!} i_{\cF}^{*}\mathrm{yo}^{*}\to \id   $ then yields the transformation 
 $\mathrm{yo}_{!}E^{\%,\cF}\to (j_{!}E)^{\infty}$.
 We finally apply $j^{*} (Y^{G})^{*}$ and get the desired transformation 
 $$j^{*}E^{\%,\cF}\to j^{*} (Y^{G})^{*} (j_{!}E)^{\infty}\simeq E \colon \bW\to \mathbf{M}\, ,$$
where the second equivalence follows form the commutativity of the   part in the diagram above.
\end{proof}

 

We let $ \bW_{\cF}^{\mathrm{hfin}}$ denote the  full subcategory of $\bW$ of spaces which are homotopy equivalent to 
a  $G$-finite $G$-CW complex with stabilizers in $\cF$.  Recall that
 $E:\bW\to \mathbf{M}$ is some functor. We call $E$ reduced if $E(\emptyset)\simeq 0$.

\begin{prop}\label{wefgbkjowgefwevffvsvfdvsfd}
Assume:
\begin{enumerate}
\item    $\bW\subseteq G\Top^{\mathrm{cf}}$  and $\bW$ contains all $G$-finite $CW$-complexes with stabilizers in $\cF$.
\item
$\bW$ is closed under taking the product with $[0,1]$.
\item $E$ is reduced, homotopy invariant, and excisive for  cell attachments.
\end{enumerate}
Then the natural transformation from Lemma \ref{sogipegrggwege}   induces an equivalence
$$(j^{*}E^{\%,\cF})_{| \bW_{\cF}^{\mathrm{hfin}}}  \to E_{| \bW_{\cF}^{\mathrm{hfin}}}\, .$$
\end{prop}
\begin{proof}   We note that $j^{*}E^{\%,\cF}:\bW\to \mathbf{M}$ is reduced, homotopy invariant, and excisive for cell attachments.

We must show that $E^{\%,\cF}(X)\to E(X)$ is an equivalence for all $X$ in  $\bW_{\cF}^{\mathrm{hfin}} $.   Since $j^{*}E^{\%,\cF}$ and $E$ are homotopy invariant we can assume that $X$ is  a $G$-finite $CW$-complexes with stabilizers in $\cF$.

 We  then argue by induction by the number of cells. 
The assertion is clear for the empty $G$-CW-complex since both functors are reduced.
 Assume now that the assertion is true for  the $G$-CW-complex $Y$ and $X$ is obtained from $Y$ by a cell-attachement. Then we have a push-out diagram
 $$\xymatrix{G/K\times S^{n}\ar[r]\ar[d]&Y\ar[d]\\G/K\times D^{n+1}\ar[r]&X}$$
 where $n$ in $\nat$ and $K$ is a subgroup of $G$ belonging to $\cF$.
 The natural transformation induces a map of push-out diagrams
 $$\xymatrix{E^{\%,\cF}(G/K\times S^{n})\ar[r]\ar[d]&E^{\%,\cF}(Y)\ar[d]\\mathbb{E}^{\%,\cF}(G/K\times D^{n+1})\ar[r]&E^{\%,\cF}(X)}\to \xymatrix{E(G/K\times S^{n})\ar[r]\ar[d]&E(Y)\ar[d]\\mathbb{E}(G/K\times D^{n+1})\ar[r]&E(X)}$$
 which is implemented by equivalences  at the two upper and the lower left corners by  the induction hypothesis.
 We conclude that
 $E^{\%,\cF}(X)\stackrel{\simeq}{\to} E(X)$.
 \end{proof}

 
We now consider two functors $E,F:\bW\to \mathbf{M}$ and assume that we are given  an equivalence
$$\phi:E_{|G_{\cF}\mathbf{Orb}}\to F_{|G_{\cF}\mathbf{Orb}}\ .$$
 \begin{kor}\label{wegtopwgwgregergw}
Assume:
\begin{enumerate}
\item    $\bW\subseteq G\Top^{\mathrm{cf}}$  and $\bW$ contains all $G$-finite $CW$-complexes with stabilizers in $\cF$.
 \item
$\bW$ is closed under taking the product with $[0,1]$.
\item $E$ and $F$ are reduced, homotopy invariant, and excisive for  cell attachments.
\end{enumerate}
Then 
$\phi$ extends to an equivalence
$$\tilde \phi:E_{|\bW_{\cF}^{\mathrm{hfin}}}\to F_{ |\bW_{\cF}^{\mathrm{hfin}}}\, .$$
\end{kor}
\begin{proof}
The equivalence $\phi$ induces an equivalence
$\tilde \phi:E^{\%,\cF}\stackrel{\simeq}{\to} F^{\%,\cF}$.
The desired equivalence is now given by the composition 
$$ E_{|\bW_{\cF}^{\mathrm{hfin}}}\stackrel{\simeq}{\leftarrow}(j^{*}E^{\%,\cF})_{|\bW_{\cF}^{\mathrm{hfin}}}\stackrel{\tilde \phi,\simeq}{\to}
(j^{*}F^{\%,\cF})_{|\bW_{\cF}^{\mathrm{hfin}}} \stackrel{\simeq}{\to}F_{|\bW_{\cF}^{\mathrm{hfin}}}$$
where the outer equivalences are supplied by Proposition \ref{wefgbkjowgefwevffvsvfdvsfd}
\end{proof}

 
 
  We let $G\mathbf{UBC}^{\mathrm{pcc},\mathrm{hfin}}$ be the full subcategory of $G\mathbf{UBC}^{\mathrm{pcc}}\cap G\Top^{\mathrm{cf}}$ of $G$-spaces which are homotopy equivalent to $G$-finite $G$-CW complexes with stabilizers in $\mathbf{Fin}$.


 \begin{prop} \label{weroigjwerogregrgwffe}We have an equivalence
$$K^{G,\cX}_{\bC}(-)_{|G\mathbf{UBC}^{\mathrm{pcc},\mathrm{hfin}}}\simeq \Sigma K\bC^{G}(-)_{|G\mathbf{UBC}^{\mathrm{pcc},\mathrm{hfin}}}\,  .$$
\end{prop}
\begin{proof}
We start with the equivalence
$$K^{G,\cX}_{\bC}(S_{min,min,disc}) \stackrel{\textrm{def}}{=}  K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(S_{min,min,disc})) \simeq \Sigma  K\bC\cX^{G}_{c,G_{can,max}}( S_{min,min})\, ,$$ where the second equivalence is given by the cone boundary  \cite[Prop.\ 9.35]{equicoarse}.
  For every $S$ in $G_{\mathbf{Fin}}\mathbf{Orb}$ the sets of invariant  locally finite subsets 
$\mathrm{LF}(S_{min,max}\otimes G_{can,min})$
and
$\mathrm{LF}(S_{min,min}\otimes G_{can,max})$ are equal.  
 Using that  $  K\bC\cX^{G}_{c}$ is a continuous equivariant coarse homology theory we get the  
middle equivalence in 
\begin{align*}
\mathclap{
K\bC\cX^{G}_{c,G_{can,max}}( (-)_{min,min})\stackrel{\textrm{def}}{=} K\bC\cX^{G}_{c}((-)_{min,min}\otimes G_{can,max})
\simeq  K\bC\cX^{G}_{c}((-)_{min,max}\otimes G_{can,min}))\stackrel{\textrm{def}}{=}  K\bC^{G}(-)
}
\end{align*}
of functors
on $G_{\mathbf{Fin}}\mathbf{Orb}$.
We now apply Corollary \ref{wegtopwgwgregergw}  with $\bW= G\mathbf{UBC}^{\mathrm{pcc}}\cap G\Top^{\mathrm{cf}} $, $\cF=\mathbf{Fin}$,
$E= K^{G,\cX}_{\bC}(-)$ and $F=\Sigma K\bC^{G}(-)$ in order to get the desired equivalence.
 \end{proof}

 



Using Proposition \ref{weroigjwerogregrgwffe} we can express  the domain of the Paschke morphism in terms of the equivariant homology theory $K\bC^{G}$.
In the following we describe the values on this functor on $G$-orbits in some detail.
 We use remark environments in order to be able to refer to this discussion later. 



\begin{rem}\label{eroigjeogiegregegegweg}
In \cite[Prop.\ 8.2.4]{coarsek} we gave an explicit description of the values of the functor $K\bC^{G}$ on $G$-orbits $S$: \begin{equation}\label{qrforfofwqewfqefqefe}
K\bC^{G}(S)\simeq K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{\mathrm{ctr}}(S_{min,max})\rtimes_{r} G)\, .
\end{equation}
Here we consider $ \mathbf{\bar C}_{\mathrm{lf}}^{\mathrm{ctr}}(S_{min,max})$ (which is the same as  ${}_{{\mathbf{K}}} \bC_{\mathrm{lf}}[\tilde S]$ in the notation of \cite{coarsek}) in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$
with the $G$-action induced by functoriality by the actions of $G$ on $S$ and $ \bC$,
and $-\rtimes_{r} G$ is the reduced crossed product for $G$-$C^{*}$-categories introduced in \cite{cank}.
Note that the objects of $ \mathbf{\bar C}_{\mathrm{lf}}^{\mathrm{ctr}}(S_{min,max})$ are objects of $\bC$ which are decomposed   as an orthogonal sums of $S$-indexed families of objects of ${\mathbf{K}}^{u}$ with finitely many non-zero terms, and morphisms are morphisms in $\bC$ which are diagonal with respect to this decomposition. 
We note that \eqref{qrforfofwqewfqefqefe} implies that the functor $K\bC^{G}$ is the functor
defined in \cite[Def.\ 17.19]{cank} and denoted there by ${K^{C^{*}\mathrm{Cat}}}^{G}_{{\mathbf{K}}^{u},r}$.

The right-hand  side of the equivalence in \eqref{qrforfofwqewfqefqefe} reflects the functorial dependence on $S$ in an obvious manner.  If one is not interested in functoriality, then  one can give even simpler formulas.  For a   subgroup
 $H$ of $G$ we have, using \cite[Prop.\ 17.18]{cank} and the Morita invariance of $K^{C^{*}\mathrm{Cat}}$, the equivalence
$$ K\bC^{G}(G/H)\stackrel{\eqref{qrforfofwqewfqefqefe}}{\simeq}  K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{\mathrm{ctr}}((G/H)_{min,max})\rtimes_{r} G)\simeq  K^{C^{*}\mathrm{Cat}}( {\mathbf{K}}^{u}\rtimes_{r} H) \, ,$$
where ${\mathbf{K}}^{u}$ is the full subcategory of ${\mathbf{K}}$ of  unital objects.
  \hspace*{\fill}$\qed$
\end{rem}



\begin{rem}\label{oiejgoigsvfdvfvs}
We continue the calculations from Remark \ref{eroigjeogiegregegegweg} but now specialize further
to the case $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}^{G}_{c}(A))$  for an $A$ in $\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$.
If $A$ is unital, then we have 
$$ K^{C^{*}\mathrm{Cat}}( {\mathbf{K}}^{u}\rtimes_{r} H)\simeq   K^{C^{*}\mathrm{Alg}}( A\rtimes_{r} H)$$
by \cite[Lem.\ 17.25]{cank}.
So in this case
$$K\bC^{G}(G/H)\simeq   K^{C^{*}\mathrm{Alg}}( A\rtimes_{r} H)\, .$$
We see that  the functor $K\bC^{G}$ has the same values as the functor introduced  in  \cite{davis_lueck}  (with additions by \cite{joachimcat}
  or alternatively  by \cite{Land:2017aa}).\footnote{To be precise, in   \cite{davis_lueck} only the case $A=\C$ is considered, but the generalization to unital $C^{*}$-algebras with trivial $G$-action is straightforward. 
  The additions concern a correction in the construction of a $K$-theory functor  for $C^{*}$-categories.}
 If $A$ is unital and {is equipped with} the trivial $G$-action, 
 then  by \cite[Prop.\ 17.24]{cank}  the functor  $K\bC^{G}$ and the Davis--L\"uck functor are  actually  equivalent as functors.  
 For general $A$ in $\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ (possibly non-unital and with non-trivial $G$-action) we propose to consider $K\bC^{G}$ as the appropriate 
 generalization of the Davis--L\"uck construction.   
 Such a generalization  was  constructed  recently also  in 
   \cite{kranz}. {If $A$ is unital and separable, then the equivalence of the two constructions is shown in Proposition \ref{weroigjoewrgergergfewrf}.}
  
\hspace*{\fill}$\qed$
\end{rem}

 

The domain and the target of the general Paschke morphism involves the $K$-theory of $C^{*}$-categories.  For general coefficients $(\bC,{\mathbf{K}})$ this can not be simplified. 
In order to get expressions involving $C^{*}$-algebras we specialize to the case \begin{equation}\label{qewfwefewfwdqewd}
(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A), \mathbf{Hilb}^{G}_{c}(A))
\end{equation}  for   $ A $ in $\Fun(BG,{C^{*}\mathbf{Alg}^{\mathrm{nu}}})$.  Note that by  Proposition \ref{qroigoergergwggrgwerg} the pair $({\mathbf{K}},\bC)$ satisfies all assumptions for Theorem \ref{qreoigjoergegqrgqerqfewf}.
Using  \eqref{rgpfojewropgegwgwrgrgwegwerg} and Proposition \ref{weroigjwerogregrgwffe} combined with Remark 
\ref{oiejgoigsvfdvfvs} we can describe the values  on the orbit category for the functor   $K^{G,\cX}_{\bC}$ appearing in the domain of the Paschke morphism.
 
 \begin{kor}
 For every subgroup $H$ of $G$ we have  an equivalence $$K^{G,\cX}_{\bC}((G/H)_{min,min,disc})\simeq \left\{\begin{array}{cc}
 0&|H|=\infty\,,
 \\\Sigma  K^{C^{*}\mathrm{Alg}}(A\rtimes_{r}H)&|H|<\infty\,.
 \end{array}\right.$$
 \end{kor}

We now turn {our} attention to the target of   the Paschke morphism. {In order to state the results properly, we introduce the following notation.}  {
\begin{ddd
\label{werigjweogwergwregwerg}\mbox{}\begin{enumerate} \item  \label{wrijgowegewrgwregwreg}We let  $G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}$ denote  the full subcategory of $G\mathbf{LCH}^{\mathrm{prop}}_{+} $ on spaces which are  homotopy equivalent in $G\mathbf{LCH}^{\mathrm{prop}}_{+} $  to   $G$-finite $G$-CW complexes with finite stabilizers. \item \label{qewfwefdwdwqedweq1} We let $G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}$ denote the full subcategory of $G\mathbf{LCH}^{\mathrm{prop}}_{+}$ of second countable spaces with proper $G$-action which are homotopy equivalent in $G\mathbf{LCH}^{\mathrm{prop}}_{+} $ to countable  $G$-CW complexes with  proper $G$-action. \end{enumerate} \end{ddd}}
  We {first show that, assuming  {$(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A), \mathbf{Hilb}^{G}_{c}(A))$,}
  we} can express the functor 
  $$K^{G,\mathrm{An}}_{\bC}(-)\stackrel{\eqref{vfdsvsvvsvsdvdsadsvdsva}}{=}\mathrm{KK}^{G}(C_{0}(-),\mathbf{Q}^{(G)}_{\mathrm{std}})$$
  in terms of the more familiar functor
 $$ K_{A}^{G,\mathrm{an}}(-) \coloneqq \mathrm{KK}^{G}(C_{0}(-),A)
$$
from $G\mathbf{LCH}^{\mathrm{prop}}_{+}$ to $\Sp$, see \cite[Def.\ 1.13]{KKG}.
 \begin{prop}\label{wegojeogrregregwergwegre}\mbox{} If $A$ is unital, then
 we have  an equivalence of functors  $$(\Sigma K_{A}^{G,\mathrm{an}})_{|G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}}\simeq  (K_{\bC}^{G,\mathrm{An}})_{|G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}}\, .$$
  \end{prop}
\begin{proof}
{Using the notation of \cite{KKG}, we have}  $K_{\mathbf{Q}^{(G)}_{\mathrm{std}}}^{G,\mathrm{an}}(-) = K^{G,\mathrm{An}}_{\bC}(-)$. 

  If $X$ is in $G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}$, then {$\mathrm{kk}^{G}(C_0(X))$ is ind-$G$-proper in the sense of \cite[Def.\ 1.24]{KKG}, and by 
Lemma \ref{eroigjweogregwgefwe}}
 the functor 
${\mathbf{B}}\mapsto K_{{\mathbf{B}}}^{G,\mathrm{an}}({X})$ sends exact sequences  in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ to fibre sequences of functors on $G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}$.
By \cite[Thm.\ 1.29.3\&7]{KKG} in combination with \cite[Thm.\ 1.25]{KKG} it furthermore
 annihilates flasques, and sends weak Morita equivalences to equivalences.
We apply  the exactness property to the exact sequence \begin{equation}\label{QEWFQWEFFQWDWD}
0\to {\mathbf{K}}^{(G)}_{\mathrm{std}}\to \bC^{(G)}_{\mathrm{std}}\stackrel{{\pi}}{\to} \mathbf{Q}^{(G)}_{\mathrm{std}}\to 0\ .
 \end{equation}
Since  $  \bC^{(G)}_{\mathrm{std}}$ is countably additive  it is    flasque and  $ K_{ \bC^{(G)}_{\mathrm{std}}}^{G,\mathrm{an}} (-)\simeq 0$.
We thus get the equivalence \begin{equation}\label{weogjkopfegwreffsv}
K_{\bC}^{G,\mathrm{An}}(-) =    K_{ \mathbf{Q}^{(G)}_{\mathrm{std}}}^{G,\mathrm{an}} (-) \stackrel{\simeq}{\to}\Sigma K_{ {\mathbf{K}}^{(G)}_{\mathrm{std}}}^{G,\mathrm{an}} (-)\end{equation} of functors on $G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}$.
  We now use that the inclusion $ A\to   {\mathbf{K}}^{(G)}_{\mathrm{std}}$ is a weak Morita equivalence \cite[Ex.\ 16.13]{cank}. It is here where we use that $A$ is unital.
   Consequently, {by \cite[Thm.\ 1.29.3]{KKG}} we get an equivalence
  \begin{equation}\label{weogjkopfegwreffsv1}K_{A}^{G,\mathrm{an}}(-)  \stackrel{\simeq}{\to}  K_{ {\mathbf{K}}^{(G)}_{\mathrm{std}}}^{G,\mathrm{an}} (-)\end{equation}
  {of functors on $G\mathbf{LCH}^{\mathrm{prop}}_{+}$.}
  Composing the equivalences in \eqref{weogjkopfegwreffsv} and \eqref{weogjkopfegwreffsv1}
  we get the asserted equivalence.
 \end{proof} 




{In the next proposition we calculate the values}
  of the functor $K_{\bC}^{G,\mathrm{An},\mathrm{lf}}$ from \eqref{regrwefefefewf}.
  We  {continue to} assume   {$(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A), \mathbf{Hilb}^{G}_{c}(A))$} 
  und use the notation introduced in Definition \ref{werigjweogwergwregwerg}.\ref{qewfwefdwdwqedweq1}.
\begin{prop} \label{werigowergrefwerfwe}
If $A$ is unital and separable, then we have an equivalence
$$(\Sigma K_{A}^{G,\mathrm{an}})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}\simeq (K_{\bC}^{G,\mathrm{An},\mathrm{lf}})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}\ .$$
\end{prop}
\begin{proof}
The argument is similar as for Proposition \ref{wegojeogrregregwergwegre}. {However,} if $X$ is in $G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}$, then 
$\mathrm{kk}^{G}(C_{0}(X))$ is not ind-$G$-proper in general so that ${\mathbf{B}}\to K^{G,\mathrm{an}}_{{\mathbf{B}}}(X)$ does not send all exact sequences to fibre sequences, i.e., Lemma \ref{eroigjweogregwgefwe} is not directly applicable.  


In analogy with \eqref{wergeefeferfewrferf} we can define 
 the  locally finite evaluation  $ F^{\mathrm{lf}}$ of 
 any functor $F$ on $G\mathbf{LCH}^{\mathrm{prop}}_{+}$  (with complete target)  
  by
$$ F^{\mathrm{lf}}(X)\coloneqq \lim_{U\subseteq X}F(U)\ ,$$
where $U$ runs over the open subsets of $X$ with $G$-compact closure.  
We have a natural transformation $c_{F}\colon F\to F^{\mathrm{lf}}$, and the transformation
$c_{F^{\mathrm{lf}}}\colon F^{\mathrm{lf}}\to (F^{\mathrm{lf}})^{\mathrm{lf}}$ is an equivalence by a cofinality argument.


We will construct an equivalence 
\begin{equation}\label{qwefrfrfrfrfrfqweoifjqwioefeqwfq1}
(\Sigma K_{A}^{G,\mathrm{an},\mathrm{lf}})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}\simeq (K_{\bC}^{G,\mathrm{An},\mathrm{lf}})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}} 
\end{equation}
and furthermore show that the canonical morphism $c_{K_{A}^{G,\mathrm{an}}}$ induces an equivalence
 \begin{equation}\label{werfwefewfqwefqwedqdqwedwer}
  (K_{A}^{G,\mathrm{an} })_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}\stackrel{\simeq}{\to}  (K_{A}^{G,\mathrm{an},\mathrm{lf}})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}\ .
  \end{equation}
 The asserted equivalence is  then defined  as the composition of the equivalences
 in \eqref{qwefrfrfrfrfrfqweoifjqwioefeqwfq1} and \eqref{werfwefewfqwefqwedqdqwedwer}.

 
We start with the construction of  \eqref{qwefrfrfrfrfrfqweoifjqwioefeqwfq1}.
We consider the following diagram in $\mathrm{KK}^{G}$
\begin{equation}\label{fqwfwefwefewfewffqewf} 
 \xymatrix{&\mathrm{kk}^{G}({\mathbf{K}}^{(G)}_{\mathrm{std}})\ar[r]\ar@{..>}[d]^{i}&\mathrm{kk}^{G}(\bC^{(G)}_{\mathrm{std}})\ar[r]^{\mathrm{kk}^{G}(\pi)}\ar@{=}[d]&\mathrm{kk}^{G}(\mathbf{Q}^{(G)}_{\mathrm{std}})\ar@{=}[d]\\\Sigma^{-1} \mathrm{kk}^{G}(\mathbf{Q}^{(G)}_{\mathrm{std}})\ar[r]^{j}&
 F(\pi)\ar[r]&\mathrm{kk}^{G}(\bC^{(G)}_{\mathrm{std}})\ar[r]^{\mathrm{kk}^{G}(\pi)}&\mathrm{kk}^{G}(\mathbf{Q}^{(G)}_{\mathrm{std}})\ .
}
 \end{equation}
 The lower part is a segment of  a fibre sequence with $F(\pi)$ defined as the fibre of $\mathrm{kk}^{G}(\pi)$.   The upper composition vanishes  since \eqref{QEWFQWEFFQWDWD} is exact, but it is not necessarily part of a fibre sequence since we only know that $\mathrm{kk}^{G}$ sends semisplit exact sequences to  fibre sequences and it is not clear that  \eqref{QEWFQWEFFQWDWD} is semisplit exact. The dotted arrow and the corresponding square is then given by the universal property of the fibre. 

  
 We  consider an ind-$G$-proper object $P$ and apply the exact  functor 
 { $\mathrm{KK}^{G}(P,-)\colon \mathrm{KK}^{G} \to \Sp$  to \eqref{fqwfwefwefewfewffqewf}}. We then  get the following diagram in $\Sp$  (as usual we drop the symbol $\mathrm{kk}^{G}$ if we insert objects in $\mathrm{KK}^{G}(-,-)$)
 \begin{equation}\label{fqwfwefwefewfewffqewf1} 
 \xymatrix{&\mathrm{KK}^{G}(P, {\mathbf{K}}^{(G)}_{\mathrm{std}})\ar[r]\ar@{..>}[d]^{i_{*}}_{\simeq}&\mathrm{KK}^{G}(P,\bC^{(G)}_{\mathrm{std}})\ar[r]^{\pi_{*}}\ar@{=}[d]&\mathrm{KK}^{G}(P,\mathbf{Q}^{(G)}_{\mathrm{std}})\ar@{=}[d]\\\Sigma^{-1} \mathrm{KK}^{G}(P,\mathbf{Q}^{(G)}_{\mathrm{std}})\ar[r]_{\simeq}^{j_{*}}&
 \mathrm{KK}^{G}(P,F(\pi))\ar[r]&\mathrm{KK}^{G}(P,\bC^{(G)}_{\mathrm{std}})\ar[r]^{\pi_{*}}&\mathrm{KK}^{G}(P,\mathbf{Q}^{(G)}_{\mathrm{std}})\ .
}
 \end{equation}
By  \cite[Thm.\ 1.29.5]{KKG} the upper sequence becomes a fibre sequence, too. Therefore the dotted arrow becomes an equivalence. Furthermore, $ \bC^{(G)}_{\mathrm{std}}$ is flasque (since it is countably additive) so that 
$\mathrm{KK}^{G}(P,\bC^{(G)}_{\mathrm{std}})\simeq 0$ by \cite[Thm.\ 1.29.7]{KKG},  and $j_{*}$ becomes  an equivalence.



 
 
  

    We consider the following two natural transformations
 \begin{equation}\label{wqefqwefqewfdqdewdqwd}
\Sigma^{-1} K_{\bC}^{G,\mathrm{An}} \stackrel{def}{=}\Sigma^{-1} K_{\mathbf{Q}^{(G)}_{\mathrm{std}}}^{G,\mathrm{an}}        \stackrel{j_{*}}{\to}  K^{G,\mathrm{an}}_{F(\pi)}
\end{equation}  
  and 
   \begin{equation}\label{wqefqwefqewfdqdewdqwd1}
K^{G,\mathrm{an}}_{A}\stackrel{\eqref{weogjkopfegwreffsv1}}{\simeq} K_{ {\mathbf{K}}^{(G)}_{\mathrm{std}}}^{G,\mathrm{an}} \stackrel{i_{*}}{\to}  K^{G,\mathrm{an}}_{F(\pi)}\end{equation}  
  of $\Sp$-valued functors on $G\mathbf{LCH}^{\mathrm{prop}}_{+}$, where $i_{*}$ and $j_{*}$ are induced by the morphisms $i$ and $j$ in \eqref{fqwfwefwefewfewffqewf}.
 
Since by \cite[Prop.\ 1.25]{KKG} the restriction of $\mathrm{kk}^{G}\circ C_{0}(-)$ to $G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}$  takes values in ind-$G$-proper objects, 
the restrictions of $j_{*}$ in \eqref{wqefqwefqewfdqdewdqwd} and     $i_{*}$  in \eqref{wqefqwefqewfdqdewdqwd1}
to $G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}$  are equivalences. 




  

 We apply the $(-)^{\mathrm{lf}}$-construction to the transformations in \eqref{wqefqwefqewfdqdewdqwd} and
\eqref{wqefqwefqewfdqdewdqwd1} and get transformations
\begin{equation}\label{qwefrfrfrfrfrfqweoifjqwioewerwerwerrfeqwfq1}
 \Sigma^{-1} K^{G,\mathrm{An},\mathrm{lf}}_{\bC} \to    K_{F(\pi)}^{G,\mathrm{an},\mathrm{lf}}
 \end{equation}
 and
  \begin{equation}\label{qwefrfrfrfrfrfqweoifjqwioefeqwfq}
  K^{G,\mathrm{an},\mathrm{lf}}_{A} \to    K_{F(\pi)}^{G,\mathrm{an},\mathrm{lf}}\ .
 \end{equation}





   
We now show that the evaluations of  \eqref{qwefrfrfrfrfrfqweoifjqwioewerwerwerrfeqwfq1}  and \eqref{qwefrfrfrfrfrfqweoifjqwioefeqwfq}    at   $X$  in $G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}$  are  equivalences. By homotopy invariance of the domains and targets   we can assume that $X$ is a countable  $G$-CW-complex with proper $G$-action.  By local compactness, it admits a cofinal family of open subsets $U$ with $G$-compact closure
belonging to $G\mathbf{LCH}^{\mathrm{prop,hfin}}_{+}$. This implies that $j_{*}$ in \eqref{wqefqwefqewfdqdewdqwd}  and $i_{*}$  in \eqref{wqefqwefqewfdqdewdqwd1}  
become equivalences after evaluation at  such $U$. We get the equivalences  \eqref{qwefrfrfrfrfrfqweoifjqwioewerwerwerrfeqwfq1}  and \eqref{qwefrfrfrfrfrfqweoifjqwioefeqwfq} as   limits of equivalences. The desired equivalence \eqref{qwefrfrfrfrfrfqweoifjqwioefeqwfq1} is now defined as the suspension of the composition  
\begin{equation}\label{fqwefwefweqdwedqe}
 (K^{G,\mathrm{an},\mathrm{lf}}_{A})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}    \stackrel{\eqref{qwefrfrfrfrfrfqweoifjqwioefeqwfq},\simeq}{\to}
  (K_{F(\pi)}^{G,\mathrm{an},\mathrm{lf}})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}\stackrel{\simeq, \eqref{qwefrfrfrfrfrfqweoifjqwioewerwerwerrfeqwfq1}}{\longleftarrow}  (\Sigma^{-1} K_{\bC}^{G,\mathrm{An},\mathrm{lf}})_{|G\mathbf{LCH}^{\mathrm{prop, \sigma hfin}}_{\mathrm{2nd},+}}  \ .
\end{equation}

  It now remains to show that the canonical  transformation
  \eqref{werfwefewfqwefqwedqdqwedwer}
      is an equivalence. 
 
We can again assume that  $X$ is a countable  $G$-CW-complex with proper $G$-action. 
 We let $(U_{n})_{n\in \nat}$ be an exhaustion of $X$ by an increasing family of invariant open  subsets with $G$-compact closure. Then setting $Y_{n}:=X\setminus  U_{n}$ the family $(Y_{n})_{n\in \nat}$ is a decreasing family of closed invariant  subsets  of $X$ with $\bigcap_{n\in \nat} Y_{n}=\emptyset$. By \cite[Thm. 1.14.3]{KKG} we get a diagram of  maps of horizontal fibre sequences 
 $$\xymatrix{\vdots\ar[d]&\vdots\ar[d]&\vdots\ar[d]\\K^{G,\mathrm{an}}_{A}(Y_{n+1}) \ar[r]\ar[d]&K^{G,\mathrm{an}}_{A}(X)\ar[d]\ar[r]&K^{G,\mathrm{an}}_{A}(U_{n+1})\ar[d]\\
 K^{G,\mathrm{an}}_{A}(Y_{n})\ar[r]\ar[d]&K^{G,\mathrm{an}}_{A}(X)\ar[r]\ar[d]&K^{G,\mathrm{an}}_{A}(U_{n})\ar[d]\\\vdots &\vdots &\vdots }\ .$$
 Here we use that the inclusions $Y_{n}\to X$ are split-closed by \cite[Prop. 5.1.1]{KKG} and our topological assumptions on $X$.
 We now consider the fibre sequence  obtained as the limit of this diagram in the vertical direction. Using that $A$ is separable and 
  \cite[Thm. 1.14.6]{KKG}  the limit of the left column vanishes. Hence
  we get an equivalence
  $$K^{G,\mathrm{an}}_{A}(X)\stackrel{\simeq}{\to} \lim_{n\in \nat} K^{G,\mathrm{an}}_{A}(U_{n+1})\simeq K_{A}^{G,\mathrm{an},\mathrm{lf}}(X)$$
  as desired.
  %

  \end{proof}



As explained already in the introduction 
the classical  definition of the domain of the Paschke morphism
does not involve a $C^{*}$-category of controlled Hilbert spaces but the choice of
a single sufficiently large  continuously controlled Hilbert space.  In order to compare the approach of the present paper with the classical one we 
  further specialize to the case of trivial coefficients characterized by 
 $$(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{(G)}(\C), \mathbf{Hilb}^{(G)}_{c}(\C))\, .$$
 According to Definition \ref{qegijqgiofjfqewfqwef} the objects of $\mathbf{Hilb}^{G}(\C)$ are pairs $(H,\rho)$ of a Hilbert space 
 $H$ and a unitary representation $\rho:G^{\mathrm{op}}\to U(H)$, $g\mapsto \rho_{g}$.
 The morphisms are given by  $\Hom_{\mathbf{Hilb}^{(G)}(\C)}((H,\rho),(H',\rho'))=B(H,H')$.
 The group $G$ fixes the objects of $\mathbf{Hilb}^{(G)}(\C)$ and acts on the morphisms
 by $g\cdot A:=\rho_{g}^{\prime, -1}A \rho_{g}$. 
 
 We consider a second countable proper metric space $X$ with an isometric action of $G$.
In the following we define an exact sequence of  $C^{*}$-categories  
 $$0\to \bC^{G}(X) \to \bD^{G}(X)\to \mathbf{Q}^{G}(X)\to 0\ .$$
 We start with the definition of a $C^{*}$-category   ${\mathbf{B}}(X)$ with $G$-action. Its objects   are triples $(H,\rho,\phi)$, where $H$ is separable and
 $\phi:C_{0}(X)\to B(H)$ is an equivariant homomorphism of $C^{*}$-algebras. Here equivariance means that 
 $g^{-1}\cdot \phi(f)=\phi(g^{*}f)$ for all $f$ in $C_{0}(X)$ and $g$ in $G$,  see \eqref{gwwgegdfsfvfdvsdvsfv}. 
 The morphisms of ${\mathbf{B}}(X)$  are inherited from $\mathbf{Hilb}^{(G)}(\C)$. The group $G$ fixes the objects of ${\mathbf{B}}(X)$ and acts on morphisms as in $ \mathbf{Hilb}^{(G)}(\C)$.
 
    
 Let $(H,\rho,\phi)$ and $ (H',\rho',\phi')$ be two objects of ${\mathbf{B}}(X)$.
  An operator $A$ in $B(H,H')$ is called locally compact  if $\phi'(f)A$ and $A\phi(f)$  belong to $K(H,H')$
 for all $f$ in $C_{0}(X)$. It is called pseudolocal if $\phi'(f)A-A\phi(f)\in K(H,H')$ for all $f$ in $C_{0}(X)$.
 Finally it is called controlled, if there exists $R$ in $(0,\infty)$ such that
 $d(\supp(f'),\supp(f))\ge R$ implies  that $\phi'(f)A\phi(f)=0$. 
 The $C^{*}$-category $\bC^{G}(X)$ is the wide subcategory of ${\mathbf{B}}(X)$ 
   generated by the invariant,  locally compact and and controlled operators. Similarly the $C^{*}$-category
   $\bD^{G}(X)$ is generated by the invariant, pseudolocal and controlled operators.
   Finally $\mathbf{Q}^{G}(X)$ is defined as the quotient. 
   If $(H,\rho,\phi)$ is an object of ${\mathbf{B}}(X)$, then the corresponding endomorphism algebras fit into an exact sequence 
   $$0\to  C(H,\rho,\phi)\to D(H,\rho,\phi)\to Q(H,\rho,\phi)\to 0$$
   which is the equivariant generalization of \eqref{wergpowkgpowergregregwegrg} from the introduction.
  \begin{ddd}\label{werogihetgergfdsf}
$(H,\rho,\phi)$ is called absorbing if for any other $(H',\rho',\phi')$ there exists  an isometry $u:(H',\rho',\phi')\to (H,\rho,\phi)$
in $\bD^{G}(X)$.
\end{ddd}
 The existence of absorbing objects in the case of trivial $G$ follows from \cite[Lem.\ 7.7]{hr}.\footnote{We {neither} know a reference {nor have a proof} for the existence of absorbing objects in the {general} equivariant case, {see Remark \ref{eiruhguiewgewrgerfwefewrf}}.}
For the following we assume that we can choose an absorbing object $(H,\rho,\phi)$.  We set $Q^{G}(X):=Q^{G}(H,\rho,\phi)$ and let $Q(H)$ be the Calkin algebra of $H$ with the induced $G$-action.
 With these choices we can define the Paschke morphism 
 $$p^{(H,\rho,\phi)}_{X}:=\mu_{X}\circ \delta_{X}:KK^{G}(\C,Q^{G}(X))\to \mathrm{KK}(C_{0}(X),Q(H))$$ as in 
\eqref{rfqwpofkjpqowefewfeqwfqef}. We   assume in addition that there exists a scale $\tau$ for $X$ such that $(X,\tau)$ belong to $G\mathbf{UBC}^{\mathrm{scl}}$.
We furthermore assume that $X$ is homotopy equivalent to a  $G$-compact $G$-CW-complex with finite stabilizers.
The following proposition asserts that  the Paschke morphism $p_{(X,\tau)}$ from Definition  \ref{troigjwrtoijgergrweg} is compatible with   $p^{(H,\rho,\phi)}_{X}$.

\begin{prop}\label{iuhugiergwergergwg}
There exists a commutative   square \begin{equation}\label{feqwfewfefqewfefqewfqe}
\xymatrix{K_{\bC}^{G,\cX}(X)\ar[r]^-{\gamma}\ar[d]^{p_{(X,\tau)}}&\mathrm{KK}(\C,Q^{G}(X))\ar[d]^{p_{X}^{(H,\rho,\phi)}}\\ K_{\bC}^{G,\mathrm{An}} (X)&\ar[l]_-{\simeq}  \mathrm{KK}^{G}(C_{0}(X) ,Q(H))  }
\end{equation}
 \end{prop}
\begin{proof}
 We use the identifications
$$K_{\bC}^{G,\cX}(X)\stackrel{\text{Lem.~}\ref{wrtoihgjoergergegwgrgwerg}}{\simeq}  \mathrm{KK}(\C,\mathbf{Q}_{\tau}(X))$$
and
$$K_{\bC}^{G,\mathrm{An}} (X)\simeq \mathrm{KK}^{G}(C_{0}(X),\mathbf{Q}^{(G)}_{\mathrm{std}})\, ,$$ see \eqref{vfdsvsvvsvsdvdsadsvdsva}.
 The objects of  $\mathbf{Q}_{\tau}(X) $ (and also of $\bD_{\tau}(X)$ and $\bC_{\tau}(X)$, see \eqref{rgfqfewfqewf1} and \eqref{rgfqfewfqewf})  are the objects of $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}_{\tau}(X)\otimes G_{can,max})$.   If $(H',\rho',\mu')$ is such an object,
 then we get the object $(H',\rho',\phi')$ of ${\mathbf{B}}(X)$  with $\phi'$ as in \eqref{ewqfpojlqkrmeflkwefqef}.  Note that  since $X$  is second countable and has the bornology relatively compact subsets, the Hilbert space $H'$ is separable by the local finiteness conditions (see Definition \ref{qeroigergeggweerg}) on  $(H',\rho',\mu')$.
Since we assume that  $(H,\rho,\phi)$ is absorbing 
 there exists  an isometry 
 $u:(H',\rho',\phi')\to (H,\rho,\phi)$ in $\bD^{G}(X)$.
 
 We consider the category $\bD^{u}_{\tau}(X) $ consisting of pairs
 $((H',\rho',\mu'),u)$ of an object  $(H',\rho',\mu')$ in $\bD_{\tau}(X)$
 and an  isometry $u$ as above. A morphism 
 $A: ((H',\rho',\mu'),u)\to ((H'',\rho'',\mu''),u')$ is a morphism
 $A: (H',\rho',\mu')\to  (H'',\rho'',\mu'')$ in $\bD_{\tau}(X)$.
 We define $\bC^{u}_{\tau}(X) $ and $\mathbf{Q}^{u}_{\tau}(X)$ similarly.
 Then we have a diagram of maps of exact sequences of $C^{*}$-categories 
 $$\xymatrix{0\ar[r]&\bC_{\tau}(X)\ar[r] &\bD_{\tau}(X)\ar[r] &\mathbf{Q}_{\tau}(X) \ar[r]&0
 \\0\ar[r]&\bC^{u}_{\tau}(X)\ar[r]\ar[d]\ar[u]&\bD^{u}_{\tau}(X)\ar[r]\ar[d]\ar[u]&\mathbf{Q}^{u}_{\tau}(X)\ar[u]\ar[d]\ar[r]&0\\ 
0\ar[r]&C^{G}(X)\ar[r]&D^{G}(X)\ar[r]&Q^{G}(X)\ar[r]&0
 }$$
 where in the lower sequence we consider the algebras as categories with a single objects. 
 The upper vertical functors  just forget the embedding $u$.
 It follows from the existence of the isometries $u$ as explained above that they are (relative) unitary equivalences.
  The definition of the lower vertical functors  on objects is clear. 
  The functor $\bD^{u}_{\tau}(X)\to D^{G}(X)$ sends a morphism $A: ((H',\rho',\mu'),u)\to ((H'',\rho'',\mu''),u')$ to $u'Au^{*}$. The other functors are defined similarly. 
  Since $K^{C^{*}\mathrm{Cat}}$ sends   (relative) equivalences to  equivalences,
  we get the diagram 
$$\xymatrix{  \ar[r]&K^{C^{*}\mathrm{Cat}}(\bC_{\tau}(X))\ar[r]\ar[d]^{\alpha}_{\simeq} &K^{C^{*}\mathrm{Cat}}(\bD_{\tau}(X))\ar[r]\ar[d] &K^{C^{*}\mathrm{Cat}}(\mathbf{Q}_{\tau}(X) )\ar[d]^{\gamma}\ar[r]& \\ 
 \ar[r]&K^{C^{*}\mathrm{Alg}}(C^{G}(X))\ar[r]&K^{C^{*}\mathrm{Alg}}(D^{G}(X))\ar[r]&K^{C^{*}\mathrm{Alg}}(Q^{G}(X))\ar[r]&
 }$$
of fibre sequences. Let left vertical map is the map $\gamma$ in   the square \eqref{feqwfewfefqewfefqewfqe}.
The map $\alpha$ is an equivalence   by \cite[Thm.\ 6.1]{indexclass}, but this will not be used here.  


If $X$ is a homotopy equivalent to $G$-finite $G$-CW complex with finite stabilizers, {then} the functor  $\mathrm{KK}^{G}(C_{0}(X),-)$ sends exact sequences in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$  to fibre sequences by a combination of \cite[Prop. 1.25]{KKG} and \cite[Thm. 1.29.5]{KKG}. 
 The lower horizontal map in  \eqref{feqwfewfefqewfefqewfqe} is induced by the functor
 $Q(H)\to \mathbf{Q}^{(G)}_{\mathrm{std}}$ which just views $(H,\rho)$ as an object of $ \mathbf{Q}^{(G)}_{\mathrm{std}}$. 
In order to show that it is an equivalence
we consider the map of fibre sequences obtained by applying $\mathrm{KK}^{G}(C_{0}(X),-)$  to the  map of exact sequences 
$$\xymatrix{ 0\ar[r]&K(H) \ar[r]\ar[d] & B(H)\ar[r]\ar[d] & Q(H)\ar[d]\ar[r]&0\\ 
0\ar[r]&{\mathbf{K}}^{(G)}_{\mathrm{std}}\ar[r]&\bC^{(G)}_{\mathrm{std}}\ar[r]&\mathbf{Q}^{(G)}_{\mathrm{std}}\ar[r]&0
 }$$
By  \cite[Cor.\ 6.18]{KKG} we    have   $\mathrm{KK}^{G}(C_{0}(X),B(H))\simeq 0 $.
We also have   $\mathrm{KK}^{G}(\C,\bC^{(G)}_{\mathrm{std}})\simeq 0 $ by  \cite[Thm.\ 1.29.7]{KKG} 
since $\bC^{(G)}_{\mathrm{std}}$   is flasque.

We claim that the left vertical map induces an equivalence after
applying $\mathrm{KK}^{G}(C_{0}(X), -)$. 
 We let $ {\mathbf{K}}^{(G),\mathrm{sep}}_{\mathrm{std}}$ and $ \bC^{(G),\mathrm{sep}}_{\mathrm{std}}$ denote the full subcategories of ${\mathbf{K}}^{(G)}_{\mathrm{std}}$  and $\bC^{(G)}_{\mathrm{std}}$, respectively, 
whose underlying Hilbert space is separable. 
Then we have a factorization
$$K(H) \to  {\mathbf{K}}^{(G),\mathrm{sep}}_{\mathrm{std}} \to {\mathbf{K}}^{(G)}_{\mathrm{std}}\ .$$
The first map is a relative Morita equivalence. In order to see this we consider the square 
$$\xymatrix{K(H)\ar[r]\ar[d]&B(H)\ar[d]\\{\mathbf{K}}^{(G),\mathrm{sep}}_{\mathrm{std}}\ar[r]& \bC^{(G),\mathrm{sep}}_{\mathrm{std}}}$$
where the horizontal maps are ideal inclusions, and 
 the right vertical map is a Morita equivalence. 
  Hence we get an equivalence 
 $$\mathrm{KK}^{G}(C_{0}(X),K(H))\stackrel{\simeq}{\to} \mathrm{KK}^{G}(C_{0}(X),{\mathbf{K}}^{(G),\mathrm{sep}}_{\mathrm{std}})$$ by \cite[Thm.\ 1.29.3]{KKG}. 
We furthermore see that $\C\to  {\mathbf{K}}^{(G),\mathrm{sep}}_{\mathrm{std}}$ and $\C\to  {\mathbf{K}}^{(G)}_{\mathrm{std}}$
are weak Morita equivalences and induce equivalences after applying $  \mathrm{KK}^{G}(C_{0}(X),-)$, again by \cite[Thm.\ 1.29.3]{KKG}. 
Hence also the inclusion $  {\mathbf{K}}^{(G),\mathrm{sep}}_{\mathrm{std}} \to {\mathbf{K}}^{(G)}_{\mathrm{std}}$ induces an equivalence after applying
this functor.
We conclude that $\mathrm{KK}^{G}(C_{0}(X),K(H))\to \mathrm{KK}^{G}(C_{0}(X), {\mathbf{K}}^{(G)}_{\mathrm{std}})$ is  an equivalence which finishes the verification of the claim.
All facts together imply  that 
$$\mathrm{KK}^{G}(C_{0}(X),Q(H))\to \mathrm{KK}^{G}(C_{0}(X), \mathbf{Q}^{(G)}_{\mathrm{std}})$$ is an equivalence, too.
 This explains the lower horizontal equivalence in \eqref{feqwfewfefqewfefqewfqe}.


 


It is now obvious from the definitions of the Paschke morphisms in \eqref{rfqwpofkjpqowefewfeqwfqef}  and Definition \ref{qeorigjoqfeqwfqfewf}   that the diagram commutes.
\end{proof}


 In the following we assume that $(X,\tau)$ satisfies the Assumptions  {of  Theorem} \ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf} such that   $p_{(X,\tau)}$  is an equivalence.
\begin{kor}\label{tgjiogweregwergre}
The morphism $\gamma$ is an equivalence if and only if $p^{(H,\rho,\phi)}_{X}$ is an equivalence. 
\end{kor}

This says that in all cases where the classical Paschke morphism $p^{(H,\rho,{\phi})}_{X}$ is an equivalence it is equivalent
to our morphism $p_{(X,\tau)}$ as a spectrum map.
An independent proof\footnote{We do not know a reference for such a proof.}  that $\gamma$ is an equivalence would allow us to conclude  from  Theorem \ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf}  that
$p^{(H,\rho,{\phi})}_{X}$ is an equivalence.

\begin{rem}\label{eiruhguiewgewrgerfwefewrf}
This is a remark about the existence of absorbing objects an in Definition \ref{werogihetgergfdsf}. First of all the discussion above depends on the existence of an absorbing object in $\bD^{G}(X)$ for which we {neither} have a reference {nor a proof}.  Related results are \cite[Lem.\ 4.5.5 \& Prop.\ 4.5.14]{willett_yu_book}. They are adapted for the approach based on localization algebras but 
do not imply directly what we need.  
A similar remark applies to   \cite[Thm.\ 1.3]{Benameur:2020aa}.

In the non-equivariant {case} the existence of  absorbing objects  is settled {e.g.\  in \cite[Lem.\ 7.7]{hr}
by an application of Voiculescu's Theorem.}

We furthermore  do not know a reference for the fact that
$p_{X}^{(H,\rho,{\phi})}$ is an equivalence. In fact,  
 \cite[Thm.\ 1.5]{Benameur:2020aa}  states a Paschke duality
 isomorphism in the equivariant case. But it is not obvious how to identify the targets and the maps {in} \cite[Thm.\ 1.5]{Benameur:2020aa} with $p_{X}^{(H,\rho,\phi)}$.
\hspace*{\fill}$\qed$
\end{rem}
 

 \section{Homotopy theoretic  and analytic assembly maps}\label{weroigjwoergergwgre}
 
 In this section we describe the homotopy theoretic and the analytic assembly maps which we will eventually compare in Theorem \ref{wtoiguwegwergergregwe}.  The homotopy theoretic assembly introduced Definition \ref{qroeigjoqergerqfewewfqewfeqwf}  is a standard construction from equivariant homotopy theory \cite{davis_lueck}.  For the historic development of the analytic assembly map we refer to \cite{Aparicio:2019aa}.
 Our Definition \ref{wergoijowergerreggwgw} is a spectrum valued refinement of the assembly map of \cite{kasparovinvent}, \cite{baum_connes_higson} which is new in this form.
 
We begin with the homotopy theoretic assembly map. Let $G\mathbf{Orb}$ denote the orbit category of $G$ and $\mathbf{M}$ be some cocomplete stable $\infty$-category. Recall that by Definition \ref{weigjorgdg}
  an equivariant $\mathbf{M}$-valued homology theory  is simply  a functor 
 $$E:G\mathbf{Orb}\to \mathbf{M}\ .$$ 
   
 Let  $\cF $ be a  family of subgroups of $G$. By $G_{\cF}\mathbf{Orb}$ we denote the full subcategory of the orbit category $G\mathbf{Orb}$ of transitive $G$-sets with stabilizers in $\cF$.  
Since 
  $*$ is a final object of $G\mathbf{Orb}$ we have a natural transformation    $E\to\underline{E(*)}$ in $\Fun( G\mathbf{Orb} , \mathbf{M})$. This transformation induces the  homotopy theoretic assembly map:
\begin{ddd}\label{woiejtgoiwegegrgergwer}The homotopy theoretic assembly map for $E$ and $\cF $  is the canonical morphism  $$\mathrm{Asmbl}^{h}_{E,\cF} \colon \colim_{G_{\cF}\mathbf{Orb}}E\to E(*)$$
  in $\mathbf{M}$.  
\end{ddd}


Recall that we can evaluate the equivariant homology theory $E$
on $G$-topological spaces using \eqref{adfiuhuihavfvasdcacsadcasca}.
For every  $X$ in $ G\Top$ we get  a morphism \begin{equation}\label{qwefeqwdqwed}
\mathrm{Asmbl}_{E,X}^{h}: E(X)\to E(*)
\end{equation}
which is induced by the projection $X\to *$.  
We let $E_{\cF}G^\mathrm{CW}$ be a $G$-CW complex representing the homotopy type of the classifying space for the family $\cF$. It is characterized 
essentially uniquely by the condition that
$$Y^{G}(E_{\cF}G^\mathrm{CW})(S)\simeq \left\{\begin{array}{cc} \emptyset&S\not\in G_{\cF}\mathbf{Orb}\,,\\{*}&S\in G_{\cF}\mathbf{Orb}\,. \end{array}\right.$$
As a consequence  of \eqref{qwefpojfopwefqwefqwefqew} we then get the equivalence
$E(E_{\cF}G^\mathrm{CW})\simeq \colim_{G_{\cF}\mathbf{Orb}}E$, and under this identification we have the equivalence \begin{equation}\label{ewfqpijoijoioij}
 \mathrm{Asmbl}_{E,\cF}^{h}\simeq   \mathrm{Asmbl}_{E,E_{\cF}G^\mathrm{CW}}^{h} 
\end{equation}
 of assembly maps.
 {Further below, in the special case of the  functor $E=K_{A}^{G}$ {which we will} introduce in Definition \ref{egiueheiugheiugwegerggwfewr} for  $A$ in $ \mathrm{KK}_{\sepa}^{G}$, and for the family $\cF=\mathbf{Fin}$, we will also use the notation $\mu^{DL}_{A}:= \mathrm{Asmbl}_{K^{G}_{A},\mathbf{Fin}}^{h}$ indicating that 
 $\mu^{DL}_{A}$ is the assembly map  introduced by Davis-L\"uck  \cite{davis_lueck}.}
 

 
We have a functor \begin{equation}\label{adfvoijqiorjvoifvafsvsdv}
\iota:G\mathbf{Orb}\to G\mathbf{BC}\, , \quad S\mapsto S_{min,max}\, ,
\end{equation} 
where $S_{min,max}$ is the $G$-set $S$ equipped with the minimal coarse structure and  the maximal bornology. 
A choice of a pair $( \bC, {\mathbf{K}})$ of $\bC$ in $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})$ and an invariant ideal $ {\mathbf{K}}$ satisfying Assumption \ref{wergoiewgewrergwrgwgwg}  gives rise  to the
equivariant coarse  $K$-homology functor 
$$K\bC\cX_{c,G_{can,min}}^{G}:G\mathbf{BC}\to \Sp^{\mathrm{}}$$
(see Definition \ref{qrogijeqoifefewfefewqffe} for $K\bC\cX^{G}_{c}$ and  Definition \ref{regoiergowregrwergwreg} for the   {twist of an}{ equivariant  coarse homology theory by an object of $G\mathbf{BC}$,   in the present case by $G_{can,min}$}). The following is the technical definition of the functor described in \eqref{asdvijqoirgfvfsdvsva}.
\begin{ddd}\label{qroeigjoqergerqfewewfqewfeqwf}
We define the functor 
$$K\bC^{G} \colon  G\mathbf{Orb} \stackrel{\iota}{\to} G\mathbf{BC} \stackrel{K\bC\cX^{G}_{c,G_{can,min}}}{\longrightarrow}  \Sp^{\mathrm{}} \, .$$
\end{ddd}
 
We now apply the definitions of assembly maps explained above  to the functor $K\bC^{G}$ in place of $E$ and introduce a shorter notation.
\begin{ddd}\label{wrtiogrgergfgs}
The homotopy theoretic assembly   map  associated to $G$, $\cF$ and $( \bC, {\mathbf{K}})$ is defined to be the map
$$\mathrm{Asmbl}_{\cF}^{h}:=\mathrm{Asmbl}_{K\bC^{G},\cF}^{h}:\colim_{G_{\cF}\mathbf{Orb}} K\bC^{G} \to K\bC^{G}(*)\, .$$
\end{ddd}

More generally, for every $X$ in $\Fun(BG,\Top)$, specializing \eqref{qwefeqwdqwed}, we have the  morphism
\begin{equation}\label{vevopjrepovervfvdfv}
\mathrm{Asmbl}_{X}^{h}:=\mathrm{Asmbl}^{h}_{K\bC^{G},X}:K\bC^{G}(X)\to K\bC^{G}(*)
\end{equation}
  induced by the projection $X\to *$.



{We now turn to the analytic assembly map  whose final definition will be stated in Definition \ref{wergoijowergerreggwgw}}.
  We start with introducing the notation for its domain.    \begin{ddd}\label{wrtoigjiowrgerferfw}
 We {denote by} $G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$   the full category  of $G\mathbf{LCH}^{\mathrm{prop}}_{+}$
  on spaces on which $G$ acts properly and cocompactly.\end{ddd}
We will  describe the analytic assembly map
$\mathrm{Asmbl}^{\mathrm{an}}_{\cF}$ associated to $  (\bC,  {\mathbf{K}})$   satisfying     Assumption \ref{wergoiewgewrergwrgwgwg}  and a family $\cF$ contained in $\mathbf{Fin}$. In analogy to \eqref{qwefeqwdqwed} we will further  describe  a natural  transformation 
$$\mathrm{Asmbl}^{\mathrm{an}} :K_{\bC}^{G,\mathrm{An}}(-)\to \underline{\Sigma \mathrm{KK}(\C, {\mathbf{K}}_{\mathrm{std}}^{(G)}\rtimes_{r}G)}
$$ of functors from $ G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$ to $\Sp^{\mathrm{}}$.
  Note that  for infinite $G$ the morphism  
 \begin{equation}\label{fwfqwefedewdedwqedqewd}
\mathrm{Asmbl}^{\mathrm{an}}_{X} :K_{\bC}^{G,\mathrm{An}}(X)\to \Sigma \mathrm{KK}(\C, {\mathbf{K}}_{\mathrm{std}}^{(G)}\rtimes_{r}G)
\end{equation} 
 can  not  simply be  induced by a map $X\to *$ since $*$ and therefore this map   are  not in the category $G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$.
  If $E_{\cF}G^\mathrm{CW}$ happens to be in $G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$, then we will have an equivalence
$\mathrm{Asmbl}^{\mathrm{an}}_{\cF}\simeq \mathrm{Asmbl}^{\mathrm{an}} _{E_{\cF}G^\mathrm{CW}}$ in analogy to \eqref{ewfqpijoijoioij}.



 
The classical definition of the analytic assembly map is based on  a construction of a    family $(\mathrm{Asmbl}^{\mathrm{an}}_{X,*})_{X\inG\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}$ of homomorphisms in $\Ab^{\Z}$
  \begin{equation}\label{efovjefoivfdvsdfvsfdv}
\mathrm{Asmbl}_{X,*}^{\mathrm{an}}: K_{\bC,*}^{G,\mathrm{An}}(X)= \mathrm{KK}_{*}^{G}(C_{0}(X),  \mathbf{Q}^{(G)}_{\mathrm{std}}  )\to  \mathrm{KK}_{*-1}(\C, {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)\ ,
\end{equation}
which implement a natural transformation
\begin{equation}\label{qewfoihioqjfiofewfqwefqwefqewfefqwef}
K_{\bC,*}^{G,\mathrm{An}}\to \underline{\mathrm{KK}_{*-1}(\C, {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}\end{equation}
of functors from $G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$ to $\Ab^{\Z}$.


\begin{rem}
The shift in the degrees come from the usage of $  \mathbf{Q}^{(G)}_{\mathrm{std}}$ instead
of $ {\mathbf{K}}^{(G)}_{\mathrm{std}} $ as coefficients for $K_{\bC}^{G,\mathrm{An}}$. This choice is more natural from the point of view of Paschke duality. \hspace*{\fill}$\qed$
\end{rem}


 In the following we describe the details of the construction of $\mathrm{Asmbl}_{X,*}^{\mathrm{an}}$
thereby lifting it to the spectrum level.
The construction has three steps. The first is an application of   functor $-\rtimes G$ from \cite[Thm. 1.15]{KKG}. The second is a pull-back along the Kasparov projection given by \eqref{wefqwfefqfqewfqfe1} below.  
The last step consists of changing target categories \eqref{wefqwfefqfqewfqfe}.


\phantomsection\label{afvoijvodvasdvsdvavdv}The following discussion will be used to get rid of the choice of cut-off functions involved in the Kasparov projection. Here we can take full advantage of the $\infty$-categorical set-up.
 We let $$\mathcal{R}:  {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}^{\mathrm{op}}\to \Set$$ be the following functor:
 \begin{enumerate}\item objects: The  functor $\mathcal{R}$ sends $X$ to the set $\mathcal{R}(X)$ of all   functions 
 $\chi$ in $C_{c}(X)$ such that 
 \begin{equation}
 \label{goqiejgoigqfqwfwefqwef}\sum_{g\in G} g^{*}\chi^{2}=1\, .\end{equation}   \item morphisms: 
 The functor $\mathcal{R}$ sends a morphism
 $f:X\to X'$ in $ G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}} $ to the map
$\mathcal{R}(f):\mathcal{R}(X')\to \mathcal{R}(X)$ which sends $\chi'$ in $\mathcal{R}(X')$ to $f^{*}\chi'$ in $\mathcal{R}(X)$. 
\end{enumerate}

For $\chi$ in $\mathcal{R}(X)$
we define the Kasparov  projection \begin{equation}\label{398z9823zf983rferwfwefwrffwr}
p_{\chi}:=\sum_{g\in G} (\chi \cdot g^{*}\chi ,g)
\end{equation}
 in $C_{0}(X)\rtimes G$. Note that the sum has finitely many non-zero terms. 

 
 If $f \colon X\to X'$ is a morphism in  
 $  {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}$ and $\chi'$ is in $\mathcal{R}(X')$, then we have the relation
 $$(f^{*}\rtimes G)(p_{\chi'}) = p_{f^{*}\chi'}\ .$$
  We interpret this construction as providing  a natural transformation of contravariant $\Set$-valued functors
 $$\mathcal{R}(-)\to \Hom_{C^{*}\mathbf{Alg}^{\mathrm{nu}}}(\C,C_{0}(-)\rtimes G)$$ on $  {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}$
 which sends $\chi$ in $\mathcal{R}(X)$ to the homomorphism
 $$\C\ni \lambda\mapsto  \lambda p_{\chi} \in C_{0}(X)\rtimes G\ .$$
 Composing with $\mathrm{kk}$  we get a natural transformation of $\mathbf{Spc}$-valued functors 
 $$\ell'\mathcal{R}(-)\to \Omega^{\infty}\mathrm{KK}(\C,C_{0}(X)\rtimes G)\ ,$$
 where $\ell':\Set\to \mathbf{Spc}$ is the canonical inclusion.
 Using the      $(\Sigma^{\infty}_{+},\Omega^{\infty})$-adjunction we can interpret the result as   a transformation \begin{equation}\label{adsckuhqiuhfiucdcdscads}
\Sigma^{\infty}_{+}\ell'\mathcal{R}(-)\to \mathrm{KK}(\C,C_{0}(-)\rtimes G)
\end{equation}
 of $\Sp^{\mathrm{}}$-valued functors.
 
 Let $E:   {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}^{\mathrm{op}}\to \mathbf{M}$ be  any functor to a cocomplete target.
We have a  functor  $$q \colon  {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}\times \Delta\to    {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}$$ which sends $(X,[n])$ to $X\times \Delta^{n}$ with the $G$-action   on the first factor.      
We define the homotopification of $E$ by 
 $$ \cH(E):=q_{!} q^{*} E \colon   {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}^{\mathrm{op}}\to \Sp^{\mathrm{}}\, ,$$   where $q^{*}$ is the pull-back along $q$ and $q_{!}$ is left-adjoint of $q^{*}$, the left Kan-extension functor. 
  The unit of the adjunction $(q_{!},q^{*})$ provides a natural transformation
$E\to \cH(E)$.
We say that $E$ is homotopy invariant if the projection $  X\times \Delta^{1}  \to X$ induces an equivalence
$E(X)\stackrel{\simeq}{\to} E(X\times \Delta^{1})$. A proof of the following lemma  {is for instance implicitly given in the proof of \cite[Lemma 7.5]{Bunke:2013uq1}} 
\begin{lem}\label{fjewoifjqowfqewqwefqwefq}
 $E$ is homotopy invariant if and only if the canonical morphism $E\to \cH(E)$ is an equivalence. \end{lem}

\begin{lem}\label{riojoijiowervervfsdvfdvsfd} The projection $\mathcal{R}\to \underline{*}$ induces
 an equivalence $\cH(\Sigma^{\infty}_{+}\ell' \mathcal{R})\simeq \underline{S}$, where $S$ is the sphere spectrum.
\end{lem}
 \begin{proof}
 By the pointwise formula for the left Kan extension $q_{!}$ we must show that the projection $\mathcal{R}\to \underline{*}$ induces natural equivalence 
 $$\colim_{[n]\in \Delta} \Sigma^{\infty}_{+}\ell'\mathcal{R}(X\times \Delta^{n})\stackrel{\simeq}{\to} S$$
 for every $X$ in $ {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}$, where the colimit is taken in $\Sp^{\mathrm{}}$.
 Since $\Sigma^{\infty}_{+}$ preserves colimits it actually suffices to show that
 $ \colim_{[n]\in \Delta} \ell'\mathcal{R}(X\times \Delta^{n})\stackrel{\simeq}{\to}  *$ in $\mathbf{Spc}$. 
 Since $*$ is a final object in $\mathbf{Spc}$ naturality is then automatic.
For a simplicial set $W$ the colimit $\colim_{\Delta}\ell' W $ is given by 
$\ell(|W|)$, where  $\ell$ is as in \eqref{adsfasdfasdfadsf} and
$|W|$ in $\Top$ is the geometric realization of $W$.  Since the  geometric realization of the total space of
a trivial Kan fibration over a point is contractible it therefore 
 suffices to show that
 the map of simplicial sets
 $\mathcal{R}(X\times \Delta^{-}) \to *$ is a trivial Kan fibration.      
 So we must show that 
 for every $n$ in $\nat$  a function 
  $\chi$ in  $\mathcal{R}(\partial \Delta^{n}\times X)$  can be extended to a function $\tilde \chi $ in $\mathcal{R}(\Delta^{n}\times X)$. 
  
  For the case $n=0$ we observe that for any $X$ in $  {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}$ we have $\mathcal{R}(X)\not=\emptyset$. For $n\ge 1$,
 using barycentric coordinates  we can write a point in $\Delta^{n}$ in the form $\sigma t$ where $\sigma$ is in $[0,1]$ and
 $t$ is   in $ \partial\Delta^{n}$. Then  an  extension of $\chi$ is given by 
 $$\tilde \chi(\sigma t,x):= \sqrt{\sigma \chi(t,x)^{2}+(1-\sigma) \chi(t_{0},x)^{2}}\ ,$$ where $t_{0}$ is the zero'th vertex of the simplex.  \end{proof}

 We now use that $ \mathrm{KK}(\C,C_{0}(-)\rtimes G)$ is a homotopy invariant $\Sp^{\mathrm{}}$-valued functor. Applying $\cH$ to \eqref{adsckuhqiuhfiucdcdscads} we get a transformation \begin{equation}\label{qewfoihiew9ufj9qweofqewfewfq}
\epsilon \colon
\underline{S}\stackrel{Lem. \ref{riojoijiowervervfsdvfdvsfd}}{\simeq} \cH({\Sigma^\infty_+}\ell'\mathcal{R})\stackrel{\cH\eqref{adsckuhqiuhfiucdcdscads}}{\to} \cH( \mathrm{KK}(\C,C_{0}(-)\rtimes G))\stackrel{\simeq, Lem \ref{fjewoifjqowfqewqwefqwefq}}{\leftarrow}  \mathrm{KK}(\C,C_{0}(-)\rtimes G)
\end{equation}
 
 

{Let $A$ be an object of $\mathrm{KK}^{G}$ and consider the functor  \cite[Def. 1.13]{KKG}.
\begin{equation}\label{qwefqwefewdewdwqdwedqewd}
K^{G,\mathrm{an}}_{A}:=\mathrm{KK}(C_{0}(-),A):G\mathbf{LCH}^{\mathrm{prop}}_{+}\to \Sp\, .
\end{equation}}We have the maximal\footnote{In the present paper  we use the convention to denote the maximal crossed product by $\rtimes$ and the reduced by $\rtimes_{r}$.} crossed product functor {$-\rtimes G$   \cite[Thm. 1.21.3]{KKG} whose action on mapping spectra   induces the following natural transformation  \begin{equation}\label{wefqwfefqfqewfqfe2}
 -\rtimes G:K_{A}^{G,\mathrm{an}}(-)=\mathrm{KK}^{G}(C_{0}(-),  {A})\to \mathrm{KK}(C_{0}(-)\rtimes G,  {A}\rtimes G)\, .
\end{equation} of functors from $G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$ to $\Sp$}. 
The composition of morphisms  in $\mathrm{KK}$ provides a  natural transformation
$$ \mathrm{KK}(\C,C_{0}(-)\rtimes G)\to \map_{\Sp^{\mathrm{}}}( \mathrm{KK}(C_{0}(-)\rtimes G, {A}\rtimes G), \mathrm{KK}(\C , {A}\rtimes G))\, .$$ We interpret its pre-composition with 
 \eqref{qewfoihiew9ufj9qweofqewfewfq}  as  a natural transformation 
\begin{equation}\label{wefqwfefqfqewfqfe1}
\epsilon^{*}: \mathrm{KK}(C_{0}(-)\rtimes G, {A}\rtimes G) \to
\underline{ \mathrm{KK}(\C , {A}\rtimes G)}
\end{equation}
of functors on $ G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$ with values in $\Sp^{\mathrm{}}$. 
{The composition of \eqref{wefqwfefqfqewfqfe2} and
\eqref{wefqwfefqfqewfqfe1} is a natural transformation \begin{equation}\label{rgfewrfwweferf}
\mu^{Kasp}_{-,A,\max}:K^{G,\mathrm{an}}_{A}(-)\to \underline{ \mathrm{KK}(\C , {A}\rtimes G)}
\end{equation} of functors from $G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$ to $\Sp$.
 We  now assume} that $\cF\subseteq \mathbf{Fin}$. 
 In general we will not have
$E_{\cF}G^\mathrm{CW}$ in $ {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}$ so that we can  not apply $K^{G,\mathrm{an}}_{A}$  or $ \mu^{Kasp}_{-,A,\max}$  to $E_{\cF}G^\mathrm{CW}$ directly. We therefore adopt the following definition.
\begin{ddd}\label{rigrgbjoirjvorgvfbdfgbfgbdfgbdfbg}
We let $$RK_{{A}}^{G,{\mathrm{an}}}\colon G\Top\to \Sp^{\mathrm{}}$$ be the left Kan extension of
$(K_{{A}}^{G,{\mathrm{an}}})_{| {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}}$ along the inclusion
$$ {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}\to G\Top\ .$$
\begin{equation}\label{}
\xymatrix{ {G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}}\ar[rr]^{K_{{A}}^{G,{\mathrm{an}}}}\ar[dr]&& \Sp^{\mathrm{}}\\& G\Top\ar@{-->}[ur]_{RK_{{A}}^{G,\mathrm{an}}}\ar@{}[u]^{\Rightarrow}&}
\end{equation}
\end{ddd}

The following efinition introduces the spectrum-valued refinement of the classical  Kasparov assembly map as introduced in \cite{kasparovinvent}, \cite{baum_connes_higson}.
 \begin{ddd}\label{wergoijowergerrrfrfrfrfeggwgw}
The  Kasparov   assembly map associated to $G$, $\cF$ and $A$
is defined as  the map $$ \mu^{Kasp}_{\cF,A,\max} \colon RK^{G,{\mathrm{an}}}_{{A}} (E_{\cF}G^\mathrm{CW}) \to   \mathrm{KK}(\C, A\rtimes G)$$ induced by
the  natural transformation in \ref{rgfewrfwweferf}.
We further define
$$ \mu^{Kasp}_{\cF,A}\colon RK^{G,{\mathrm{an}}}_{{A}} (E_{\cF}G^\mathrm{CW}) \to   \mathrm{KK}(\C, A\rtimes_{r} G)$$
as the composition of $ \mu^{Kasp}_{\cF,A,\max}$ with the canonical morphism
$A\rtimes G\to A\rtimes_{r}G$.
  \end{ddd}
  Note that both versions of the Kasparov assembly map are, by construction, natural in the coefficient object $A$ in $\mathrm{KK}^{G}$.


{We now derive the analytic assembly map \eqref{fwfqwefedewdedwqedqewd} associated to a pair $(\bC,{\mathbf{K}})$.}
The composition of the transformations $-\rtimes G\to -\rtimes_{r}G$ and 
$\id\to \mathrm{Idem}$ of functors from $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ to $C^{*}\mathbf{Cat}^{\mathrm{nu}}$ yields a morphism of exact sequences.
\begin{equation}\label{rfreffewqefewfwefewffwfwfwefewfewfewfwfwefewfewfqfdfdfwefqwef} 
\xymatrix{0\ar[r]&
  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes G \ar[d] \ar[r] &  \bC^{(G)}_{\mathrm{std}}\rtimes G \ar[d] \ar[r]&   \mathbf{Q}^{(G)}_{\mathrm{std}}\rtimes G \ar[d]^{\mathrm{ctc}} \ar[r]&0\\0\ar[r]& \mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)\ar[r]& \mathrm{Idem} ( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)\ar[r]& \frac{\mathrm{Idem}(  \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}   \ar[r]&0}
\end{equation}
In the upper line we also used that the functor $-\rtimes G$ is exact.
The functor $\mathrm{ctc}$  will be called the change of target categories functor.


The change of target categories functor $\mathrm{ctc}$ in \eqref{rfreffewqefewfwefewffwfwfwefewfewfewfwfwefewfewfqfdfdfwefqwef} yields the first morphism in the following composition. The second is  the boundary map associated to the second exact sequence in \eqref{rfreffewqefewfwefewffwfwfwefewfewfewfwfwefewfewfqfdfdfwefqwef}.
Finally the last equivalence is the Moria invariance of $ \mathrm{KK}(\C,-)= K^{C^{*}\mathrm{Cat}}$ \cite[Thm. 14.14]{cank}. \begin{align}
\mathrm{KK}(\C,\mathbf{Q}^{(G)}_{\mathrm{std}}\rtimes G) & \xrightarrow{\mathrm{ctc}}  \mathrm{KK}(\C,  \frac{\mathrm{Idem}(  \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)} )\label{wefqwfefqfqewfqfe}\\
& \xrightarrow{\phantom{\mathrm{ctc}}}  \Sigma \mathrm{KK}(\C, \mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G))\xleftarrow{\simeq}\Sigma \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)\notag
\end{align}
 
{We  now specialize to $A=\mathrm{kk}^{G}(\mathbf{Q}_{\mathrm{std}}^{(G)})$, but we will
drop the symbol $\mathrm{kk}^{G}$   in order to shorten the formulas. We use that $K^{G,\mathrm{an}}_{\mathbf{Q}^{(G)}_{\mathrm{std}}}= K^{G,\mathrm{An}}_{\bC}$, compare \eqref{qwefqwefewdewdwqdwedqewd} and \eqref{vfdsvsvvsvsdvdsadsvdsva}.}
\begin{ddd} \label{qeriughioergewgergwer9}
We define the natural transformation
{$$\mathrm{Asmbl}^{\mathrm{an}}_{-} \colon K_{\bC}^{G,\mathrm{An}}(-) \stackrel{\mu^{Kasp}_{-,\mathbf{Q}_{\mathrm{std}}^{(G)},\max}}{\to}
 \underline{  \mathrm{KK}(\C,  \mathbf{Q}_{\mathrm{std}}^{(G)}\rtimes  G)}\stackrel{\eqref{wefqwfefqfqewfqfe}}{\to}
 \underline{\Sigma \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}$$ 
of functors from $ G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$ to $\Sp^{\mathrm{}}$.}  
We then define $\mathrm{Asmbl}^{\mathrm{an}}_{X,*}:=\pi_{*}(\mathrm{Asmbl}^{an}_{X})$.
\end{ddd}


 
 We now use Definition \ref{rigrgbjoirjvorgvfbdfgbfgbdfgbdfbg} for $RK^{G,\mathrm{An}}_{\bC}=
 RK^{G,\mathrm{an}}_{\mathbf{Q}^{(G)}_{\mathrm{std}}}$. 
     \begin{ddd}\label{wergoijowergerreggwgw}
The analytic assembly map associated to $G$, $\cF$ and $( \bC, {\mathbf{K}})$
is defined as  the map $$\mathrm{Asmbl}^{\mathrm{an}}_{\cF} \colon RK^{G,\mathrm{An}}_{\bC} (E_{\cF}G^\mathrm{CW}) \to  \Sigma \mathrm{KK}(\C, {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)$$ induced by  {the natural  transformation} $\mathrm{Asmbl}^{\mathrm{an}}$ in Definition \ref{qeriughioergewgergwer9}.
 \end{ddd}


\section{\texorpdfstring{$\boldsymbol{C^{*}}$}{Cstar}-categorical model for the homotopy theoretic assembly map}
 


The homotopy theoretic assembly map $\mathrm{Asmbl}_{\cF}^{h}$ in  Definition \ref{wrtiogrgergfgs} is defined  
  in terms of the homology theory  $K\bC^{G}$.
On the other hand, the analytic assembly map $\mathrm{Asmbl}^{\mathrm{an}}_{X}$ is constructed in Definition \ref{wergoijowergerreggwgw} in terms of $KK$-theory.  Our   goal is to compare these two assembly maps.  As a first step, in this section we  will construct  an assembly map $\mathrm{Asmbl}_{X}^{\Theta}$ 
induced by an explicit  functor $\Theta_{X}$ between $C^{*}$-categories and show that it is equivalent to the homotopy theoretic assembly map $\mathrm{Asmbl}^{h}_{X}$ on $G$-finite $G$-simplicial complexes. 

We assume that $( \bC, {\mathbf{K}})$ satisfies Assumption \ref{wergoiewgewrergwrgwgwg}.
 Recall Definition \ref{wtogjoergrwegreggwege} of the functor $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}$ and Definition \ref{qeroighjoreqgreqfefwfqwef} of the $G$-$C^{*}$-category $\bC^{(G)}_{\mathrm{std}}$. 
 
 

 
  Let $X$ be in $G\mathbf{BC}$. For an object $(C,\rho,\mu)$ in $    \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,min})$ we use the abbreviation  \begin{equation}\label{}
\mu_{g}:=\mu(X\otimes \{g\})
\end{equation}
denoting a projection on $C$. 
 Note that the objects of $\bC_{\mathrm{std}}^{(G)}\rtimes_{r}G$ (we refer to \cite[Def. 10. 8]{cank} for  the definition of the reduced crossed product) are the objects of $\bC_{\mathrm{std}}^{(G)}$. The latter  are pairs $(C,\rho)$ as described in   Definition \ref{tgijrogwegweffsfv}. 
 Therefore objects in $ \mathrm{Idem}(\bC_{\mathrm{std}}^{(G)}\rtimes_{r}G)$ are triples $(C,\rho,p)$, where $p$ is a projection on $(C,\rho)$ in $ \bC_{\mathrm{std}}^{(G)}\rtimes_{r}G$.  
 We refer to Proposition \ref{weoigjweogergrrewgwregwrg} below for  the verifications 
 related with the following definition.
 \begin{ddd}\label{eoigjoeirgrwegreregrgewrg}
 We define a functor   
$$\Theta_X:    \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,min})\to \mathrm{Idem}(\bC_{\mathrm{std}}^{(G)}\rtimes_{r}G)\ .$$
as follows:
\begin{enumerate}
\item objects: The functor $\Theta_X$ sends the object $(C,\rho,\mu)$ in $   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,min})$ to the object $(C,\rho,p)$ in  $\mathrm{Idem}(\bC_{\mathrm{std}}^{(G)}\rtimes_{r}G)$, where \begin{equation}\label{qewfeqwfpokpoqwefqwefqewf}
  \tilde p:=( \mu_{e},e)\, .
\end{equation} 

\item morphisms:
The functor $\Theta_{X}$ sends the morphism $A:(C,\rho,\mu)\to (C',\rho',\mu')$  in $   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,min})$ to the morphism \begin{equation}\label{ewqifhwqeifewfewfwefwefqwef}
\Theta_{X}(A):=\sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g):(C,\rho, \tilde p)\to (C',\rho', \tilde p')
\end{equation}  in $ \mathrm{Idem}( \bC_{\mathrm{std}}^{(G)}\rtimes_{r}G)$. 
 \end{enumerate}
 \end{ddd}
 
 Let $G\mathbf{BC}_{\mathrm{bd}}$ denote the full category of $G\mathbf{BC}$ of bounded $G$-bornological coarse spaces. Recall the definition of a relative idempotent completion of an ideal from \cite{cank} (this is relevant for Assertion \ref{weoigjweogergrrewgwregwrg}.\ref{ewtgojkropgwergregergwerg2} below since $ {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G$ is not unital in general).
 \begin{prop}\label{weoigjweogergrrewgwregwrg}\mbox{}
 \begin{enumerate}
 \item\label{ewtgojkropgwergregergwerg} The functor $\Theta_{X}$ is well-defined.
 \item  \label{ewtgojkropgwergregergwerg1} The family $(\Theta_{X})_{X\in G\mathbf{BC}}$ is a natural transformation \begin{equation}\label{grpogkprewgref}
\Theta:  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(-\otimes G_{can,min})\to \underline{ \mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)}
\end{equation}
of functors from $G\mathbf{BC}$ to $C^{*}\mathbf{Cat}^{\mathrm{nu}}$.
\item \label{ewtgojkropgwergregergwerg2} The transformation $\Theta$ restricts to a transformation  \begin{equation}\label{grpogkpfrfrfrfrfrewgref}
\Theta:  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(-\otimes G_{can,min})\to \underline{\mathrm{Idem}({ {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}} \end{equation}
of functors from $G\mathbf{BC}_{\mathrm{bd}}$ to $C^{*}\mathbf{Cat}^{\mathrm{nu}}$.

 \end{enumerate}
 \end{prop}
  \begin{proof}
 We start with Assertion  \ref{weoigjweogergrrewgwregwrg}.\ref{ewtgojkropgwergregergwerg}. 
  Since $X\times G$ is a free $G$-set and $(C,\rho,\mu)$ is locally finite it follows that
$(C,\rho)$ belongs to $\bC^{(G)}_{\mathrm{std}}$. 
Furthermore, it is   obvious that  $\tilde p$ 
 in \eqref{qewfeqwfpokpoqwefqwefqewf} is an orthogonal projection on the object  $(C,\rho)$ in $ \bC_{\mathrm{std}}^{(G)}\rtimes_{r}G$. Consequently, $(C,\rho,\tilde p)$ is a well-defined object in  $\mathrm{Idem}(  \bC_{\mathrm{std}}^{(G)}\rtimes_{r}G)$.

\begin{lem} \label{rgoijerwoggwergregwgreg}
The sum \eqref{ewqifhwqeifewfewfwefwefqwef} converges in $  \bC_{\mathrm{std}}^{(G)}\rtimes_{r}G$ and defines a continuous map on morphism spaces.
\end{lem}
\begin{proof}  We first observe that if $A$ has controlled propagation
then this sum has finitely many non-zero terms. We now consider 
  the defining representation  \cite[Rem. 10.5]{cank}
$$\sigma: \bC_{\mathrm{std}}^{(G)}\rtimes_{r}G\to \mathbf{L}^{2}(G,\bC^{(G)}_{\mathrm{std}})\ .$$
Note that
$\sigma(C,\rho)  =(C,\rho)$, and  for a morphism $f:(C,\rho)\to (C',\rho')$ we have $$\sigma(f,g)=\sum_{l\in G} e'_{\ell} (lg)\cdot f e_{lg}^{*}:\bigoplus_{g\in G}C\to \bigoplus_{g\in G} C'$$ by  \cite[(10.7)]{cank},
where $( \bigoplus_{g\in G} C, (e_{l})_{l\in G})$   and 
$( \bigoplus_{g\in G} C', (e'_{l})_{l\in G})$
represent the orthogonal sums of the constant families  $(C)_{g\in G}$  $(C')_{g\in G}$. 
In order to understand  that this formula is really an instance of    \cite[(10.7)]{cank}, note that $G$ fixes the objects of $\bC^{(G)}_{\mathrm{std}}$. It follows from Definition \ref{qeroigergeggweerg}.\ref{qeroighjoergfqewfqf} and \cite[Lem. 4.15]{cank} that 
 $C$ is isomorphic to the orthogonal sum of the images of the family of projections  $(\mu_{g})_{g\in G}$.   We therefore get  an isometric embedding 
$$u:C \to  \bigoplus_{g\in G} C\, , \quad u:=\sum_{g\in G} e_{g}\mu_{g}  \, .$$
We have an analogous embedding $u': C' \to  \bigoplus_{g\in G} C'$.
We still assume that $A$ is controlled. We calculate using \eqref{wfoizeqwfu98u9e8wfqewf} (saying that $g\cdot \mu_{h}=\mu_{gh}$ for all $g,h$ in $G$) and the  $G$ invariance of $A$ reflected by $g\cdot A=A$  for all $g$ in $G$ that 
\begin{eqnarray} 
\sigma(\sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)) &=&
\sum_{l,g\in G }e'_{l} (lg)\cdot   ( \mu'_{g^{-1}}  A \mu_{e} )  e_{lg}^{*}\label{wthoigwjopgwrregw}\\&=&
\sum_{l,g\in G }e'_{l}    \mu'_{l}   A \mu_{lg}   e_{lg}^{*}\nonumber\\&=&
  u' A  u^{*} \nonumber
\end{eqnarray}
This formula shows that  the right-hand side in \eqref{ewqifhwqeifewfewfwefwefqwef}
extends by continuity to all morphisms in  $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,min})$.
\end{proof}
In the following calculations we use the formulas from \cite[Def. 5.1]{crosscat} for the 
composition and adjoints in crossed products.
\begin{lem}\label{w45opktrpohwrgwreg}
We have the identities $$\tilde p \sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)=\sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)\tilde p=\sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)\, .$$
\end{lem}
 \begin{proof}

\begin{eqnarray*}
\tilde p \sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)&=&
\sum_{g\in G}  (\mu_{e},e) (    \mu'_{g^{-1}}  A \mu_{e}   ,g)\\&=&
\sum_{g\in G}  ((g^{-1}\cdot \mu_{e})   \mu'_{g^{-1}}  A \mu_{e}   ,g)\\&=&
\sum_{g\in G}  (  \mu'_{g^{-1}}  A \mu_{e}   ,g) \\
  \sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)\tilde p &=&
 \sum_{g\in G}   (    \mu'_{g^{-1}}  A \mu_{e}   ,g) (\mu_{e},e) \\&=&
\sum_{g\in G}   (    \mu'_{g^{-1}}  A \mu_{e}   ,g) \ . \end{eqnarray*}
\end{proof}
 
Lemma \ref{rgoijerwoggwergregwgreg} together with Lemma  \ref{w45opktrpohwrgwreg} shows that $  \sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)$ is a well-defined morphism 
$\Theta_{X}(A):(C,\rho,\tilde p)\to (C',\rho',\tilde p')$ in $\mathrm{Idem}(\tilde \bC_{\mathrm{std}}^{(G)}\rtimes_{r}G)$.

 \begin{lem}
$ \Theta_{X}$  is compatible with the composition and the involution.
 \end{lem}
 \begin{proof}
  Let $A':(C',\rho',\mu')\to (C'',\rho'',\mu'')$
 be a second morphism in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,min})$.
 Since $\Theta_{X}$ is known to be continuous by Lemma \ref{rgoijerwoggwergregwgreg} we can assume for simplicity that the morphisms are controlled so that all sums are finite. Then
 \begin{eqnarray*}
\Theta_{X}(A')\Theta_{X}(A)&=& \sum_{g,h\in G} (    \mu''_{g^{-1}}  A' \mu'_{e}   ,g) (    \mu'_{h^{-1}}  A \mu_{e}   ,h)\\&=&
 \sum_{g,h\in G} (  h^{-1}\cdot(  \mu''_{g^{-1}}  A' \mu'_{e} )     \mu'_{h^{-1}}  A \mu_{e}   ,gh)\\&=& 
 \sum_{g,h\in G} (    \mu''_{h^{-1}g^{-1}}  A' \mu'_{h^{-1}}       A \mu_{e}   ,gh)\\&=& 
\sum_{g,h\in G} (    \mu''_{g^{-1}}  A' \mu'_{h}       A \mu_{e}   ,g)\\&=& \sum_{g,h\in G} (    \mu''_{g^{-1}}  A'      A \mu_{e}   ,g)\\&=&
\Theta_{X}(A'A)\,.
\end{eqnarray*}

Furthermore note that
\[\Theta_{X}(A)^{*}= \sum_{g\in G} (    \mu'_{g^{-1}}  A \mu_{e}   ,g)^{*}=
 \sum_{g\in G} (   g\cdot (\mu'_{g^{-1}}  A\mu_{e})^{*}  ,g^{-1})= \sum_{g\in G} (     \mu_{g}  A^{*} \mu'_{e}    ,g^{-1})=\Theta_{X}(A^{*})\, .\qedhere\]
 \end{proof}
 
 This finishes the verification that  $\Theta_{X}$ is a well-defined functor between $C^{*}$-categories and therefore the proof of  Assertion  \ref{weoigjweogergrrewgwregwrg}.\ref{ewtgojkropgwergregergwerg}. 
 
 
 In order to see Assertion  \ref{weoigjweogergrrewgwregwrg}.\ref{ewtgojkropgwergregergwerg1} we consider a map
 $f \colon X\to X'$ in $G\mathbf{BC}$. Then $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(f)(C,\rho,\mu)=(C,\rho,f_{*} \mu)$.
 We observe by inspection of the definitions  that
 $$\Theta _{X'}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(f)(C,\rho,\mu))=\Theta_{X}(C,\rho,\mu)\ , \quad \Theta_{X}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(f)(A))=\Theta_{X}(A)\ .$$
 
 We finally show  Assertion  \ref{weoigjweogergrrewgwregwrg}.\ref{ewtgojkropgwergregergwerg2}. 
 If $X$ is bounded, then
 $X\times \{g\}$ is a bounded subset of $X\otimes G_{can,min}$ for every $g$ in $G$.
 Consequently,
 $\mu_{g}$ belongs to ${\mathbf{K}}$, see the explanations in  Remark \ref{rqfqiofioewjeowfewfqewfqwefe}. 
 In view of
 \eqref{ewqifhwqeifewfewfwefwefqwef} this implies that $\Theta_{X}$ takes values in
 $\mathrm{Idem}({\mathbf{K}}_{\mathrm{std}}^{(G)}\rtimes_{r}G)$
 \end{proof}

 


For $X$ in $G\mathbf{UBC}_{\mathrm{bd}}$ we will also write 
\begin{equation}\label{fqoifoifjoeiwfjqowfqwefqwefqwefq}
\Theta_{X}:  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes G_{can,min})\to \underline \mathrm{Idem}({ {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}
\end{equation}
for the  restriction of the functor $\Theta_{{\mathcal{O}}(X)}$ to the ideal $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes G_{can,min})$ see \eqref{f23r8z89fz2893zf892334234f2} for the notation.

We now apply the functor $K^{C^{*}\mathrm{Cat}}(-)=\mathrm{KK}(\C,-)$ to the transformations \eqref{grpogkprewgref} and \eqref{grpogkpfrfrfrfrfrewgref}. 
Using the    
  Definition \ref{qrogijeqoifefewfefewqffe}  of  $K\bC\cX^{G}_{c}$ in order to rewrite the domain and the Morita invariance of $K^{C^{*}\mathrm{Cat}}(-)$   in order to remove $\mathrm{Idem}(-)$  in the target we get the assertions of the following corollary.
\begin{kor}\label{wopijgowergerfwerfwerfewrf}\mbox{} 
\begin{enumerate} \item 
We have a natural transformation
\begin{equation}\label{}
\theta:K\bC\cX^{G}_{c,G_{can,min}} \to \underline{\mathrm{KK}(\C, \bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)}
\end{equation} of functors from $G\mathbf{BC}$ to $\Sp^{\mathrm{}}$. \item \label{wegiojreogewrfwerffw}
We have a natural transformation
\begin{equation}\label{}
\theta:K\bC\cX^{G}_{c,G_{can,min}} \to \underline{\mathrm{KK}(\C, {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}
\end{equation} of functors from $G\mathbf{BC}_{\mathrm{bd}}$ to $\Sp^{\mathrm{}}$.
\end{enumerate}
\end{kor}


 \begin{prop} \label{wrtoihjwogregergwergwreg}If  ${\mathbf{K}}$ is compatible with orthogonal sums  (Definition \ref{wefiqrjhfoifewqdqwedqwedqewd}), then   
  the  morphism  \begin{equation}\label{qwefweofjwpoefeqfeqwfqwe}
 \theta_{*}: K\bC\cX^{G}_{c,G_{can,min}}(*)\to \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)
\end{equation}
  is an equivalence.
\end{prop} \begin{proof}
The proof is very similar to the proof of Proposition \ref{4oijotrherhrthrhe}. But the difference is that here $G$ is infinite while there $H$ was finite. 
  By definition the morphism  in question is 
$$ K \bC\cX^{G}_{c,G_{can,min}}(*)\simeq K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min}))\stackrel{K^{C^{*}\mathrm{Cat}}(\Theta_{*})}{\to} K^{C^{*}\mathrm{Cat}}(\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)) 
\ .$$

We will   construct a factorization of $\Theta_{*}$  as
$$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min}) \to \bD\to \mathrm{Idem}(\bD)\to \mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)\ ,$$ where  $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min}) \to \bD$ is a weak Morita equivalence, $\bD\to \mathrm{Idem}(\bD)$ is an idempotent completion of an ideal, and $\mathrm{Idem}(\bD)\to  
 \mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)$   is a relative equivalence. Since $K^{C^{*}\mathrm{Cat}}$ sends   functors with  any of these properties to equivalences \cite[Sec. 14 -16]{cank} the assertion then follows.
 
  

 

\begin{lem}\label{reoithgjowrtegwrgwergreg}
 $\Theta_{*}$ is fully faithful.
 \end{lem}
 \begin{proof}
 It is clear from \eqref{wthoigwjopgwrregw} that $\Theta_{*}$  is injective on morphisms. It remains to show that it is surjective.
 
  Let $(C,\rho,\mu)$ and $(C',\rho',\mu')$ be two objects of $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min})$.
  Note that $\Theta_{*}(C,\rho,\mu)=(C,\rho,\tilde p)$ and $\Theta_{*}(C',\rho',\mu')=(C'\rho',\tilde p')$ in $\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)$.
 Let $A:(C,\rho,\tilde p) \to (C'\rho',\tilde p')$ be a morphism  in $\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)$. We will construct a morphism $\hat A:(C,\rho,\mu)\to (C',\rho',\mu')$ in 
 $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min})$ such that $\Theta_{*}(\hat A)=A$.
 
 Note that $A$ is a morphism $(C,\rho)\to (C',\rho')$ in $ {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G  $ which in addition satisfies   $ \tilde p' A\tilde p=A$.  
 We set $A=\sum_{g\in G} (A_{g},g)$ with $A_{g}:C\to C$ and calculate
  \begin{eqnarray*}
\tilde p' A\tilde p&=&\sum_{g\in G}   (\mu'_{e},e) (A_{g},g)   (\mu_{e},e)\\
&=& \sum_{g\in G}   (\mu'_{g^{-1}} A_{g}\mu_{e},g)  \ .
\end{eqnarray*}
Consequently,  we have 
$\mu'_{g^{-1}} A_{g}\mu_{e}=A_{g}$ for all $g$ in $G$.
 We define $$\tilde A:=\sum_{g\in G} \mu'_{g^{-1}} A_{g}\mu_{e}:C\to C'\ .$$
The calculation \eqref{wthoigwjopgwrregw} shows that the covergence of the sum 
$\sum_{g\in G} (A_{g},g)$ in norm of  $ {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G$ is equivalent to the convergence of the sum $\sum_{g\in G} \mu'_{g^{-1}} A_{g}\mu_{e}$.

Next we  want to define a morphism $$\hat A:=\sum_{h\in G} h\cdot \tilde A:C\to C'\ .$$  Using the fact that $C$ is the orthogonal sum of   the images of $(\mu_{g})_{g\in G}$ and $C'$  is the orthogonal sum of   the images of $(\mu'_{g})_{g\in G}$  we argue that $\hat A$ is a well-defined morphism (the sum does not converge in norm, but must be interpreted in the sense of \cite[Cor. 4.1]{cank})
Let $F$ be any finite subset of $G$. Then we have 
 \begin{eqnarray*}
\sum_{h\in F} (h\cdot \tilde A)^{*} (h\cdot \tilde A')&=&\sum_{h\in F}  \sum_{g,l\in G} 
h\cdot  (\mu'_{g^{-1}} A_{g}\mu_{e}  )^{*} h\cdot  (\mu'_{l^{-1}} A_{l}\mu_{e})\\&=&\sum_{h\in F}  \sum_{g,l\in G} 
 \mu_{h}   h\cdot(A_{g})^{*}    \mu'_{hg^{-1}}   \mu'_{hl^{-1}} h\cdot(A_{l}) \mu_{h}\\&=&\sum_{h\in F}  \sum_{g\in G} 
 \mu_{h}   h\cdot(A_{g})^{*}     \mu'_{hg^{-1}}   h\cdot(A_{g}) \mu_{h}\\&=&
 \sum_{h\in F}  \mu_{h} h\cdot   \left( \sum_{g\in G} 
   A_{g}^{*}     \mu'_{g^{-1}}    A_{g}\right) \mu_{h} 
\end{eqnarray*}
 Because of  $$ \sum_{g\in G} 
   A_{g}^{*}     \mu'_{g^{-1}}    A_{g}=
    \sum_{g,h\in G} 
 ( \mu_{e} A_{h}^{*}  \mu'_{h^{-1}})   (\mu'_{g^{-1}}    A_{g}\mu_{e})
  =  ( \sum_{h\in G} 
    \mu_{e} A_{h}^{*}  \mu'_{h^{-1}})^{*} (\sum_{g\in G} 
  \mu'_{g^{-1}}    A_{g}\mu_{e} )$$ we  know that $ \sum_{g\in G} 
   A_{g}^{*}     \mu'_{g^{-1}}    A_{g} $   converges  in norm to some morphism $B:C\to C$. But since the family $(\mu_{g})_{g\in G}$ is mutually orthogonal, for every morphism
   $B:C\to C$ we have 
   $ \|
 \sum_{h\in F}  \mu_{h} h\cdot  B \mu_{h} \|\le \|B\|$.
 Since the bound does not depend on the finite subset $F$ of $G$ we conclude that that $\hat A$ exists by \cite[Cor. 4.1]{cank}. 
 
 By definition we have $g\cdot \hat A=\hat A$ for all $g$ in $G$.  
 The family $(A_{g})_{g\in G}$ has  finitely many non-zero members if an only 
 if $\hat A$ is controlled as a morphism from $(C,\rho,\mu)$ to $(C',\rho',\mu')$. This implies in the general case that $\hat A$ 
 can be approximated by invariant controlled morphisms, i.e., $\hat A\in  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min})$. 
 By construction we have $\Theta_{*}(\hat A)=A$.
 This finishes the verification that $\Theta_{*}$ is full faithful. \end{proof}
 
 
  For every free $G$-set $Y$,   
  every  subset $F$ of $Y$, and  every object $(C,\rho,\mu)$ in ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(Y_{min})$
   we can consider the projection
$\tilde p_{F}:=(\mu(F),e)$ on $(C,\rho)$ considered as an object of  $\bC^{(G)}_{\mathrm{std}}\rtimes G$.
We let $\bD$ be the full subcategory of
$\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes G)$  (the idempotent completion of ${\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes G$ relative to $\bC^{(G)}_{\mathrm{std}}\rtimes G$) of objects of the form
$(C,\rho,\tilde p_{F})$ for some choice of $Y$, $F$ and $(C,\rho,\mu)$ as above. 
We can consider $Y=G$ and $F=\{e\}$. Then
$\tilde p=\tilde p_{\{e\}}$ so that $\bD$ contains the image of $\Theta_{*}$. 

Recall the notion of a weak Morita equivalence from \cite[Def. 16.2]{cank}.
\begin{lem} The functor
$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min})\to \bD$ is a weak Morita equivalence.
\end{lem}
\begin{proof}
First of all note that $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min})$ is unital.
It follows from Lemma \ref{reoithgjowrtegwrgwergreg} that  the morphism in question is fully faithful. It remains to show that 
set of objects $\Theta_{*}(\mathrm{Ob}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min})))$ is weakly generating \cite[Def. 16.1]{cank}.
In order to simplify the notation we write $\tilde p_{g}:=\tilde p_{\{g\}}$ and note that $\tilde p=\tilde p_{e}$.
We have
$$(\id_{C},g)\tilde p_{e} (\id_{C},g)^{*}=\tilde p_{g}\ .$$
This shows that for every $g$ in $G$ and $(C,\rho,\mu)$ in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(G_{can,min})
$ the  object 
$(C,\rho,\tilde p_{g})$ in $\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)$ is isomorphic to the object  $(C,\rho,\tilde p)$  
  which belongs to the image of $\Theta_{*}$.
Furthermore, 
for every finite subset $F$  of some free $G$-set $Y$
and $(C,\rho,\mu)$ in  ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(Y_{min})$   the object
$(C,\rho,\tilde p_{F})$   is isomorphic to a finite sum of objects in the image of $\Theta_{*}$.

Let now $(C,\rho,\tilde p_{F})$ be any object of $\bD$ and 
$(A_{i})_{i\in I}$ be a finite family of morphisms in $\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes G) $ with target $(C,\rho,\tilde p_{F})$.  Let $\epsilon$ be in $(0,\infty)$. 
We write $A_{i}=\sum_{h\in G} (A_{i,h},h)$ where $A_{i,h}$ belong to ${\mathbf{K}}$. Since these sums converge in norm and $I$ is finite 
  there exists a finite subset $F'$ of $G$ such that
$\|A_{i}- \sum_{h\in F'} (A_{i,h},h)\|\le \epsilon/2$ for all $i$ in $I$.
Since ${\mathbf{K}}$ is compatible with orthogonal sums we can find a finite subset $F''$ of $Y$ such that $$\|A_{i,g}-\mu(g^{-1}F'')A_{i,g}\| \le \frac{\epsilon}{2|F'| }$$ for all $i$ in $I$ and $g$ in $F'$. 
Then
$\|A_{i}-\tilde p_{F''}A_{i}\|\le \epsilon$ for all $i$ in $I$.
\end{proof}


 Recall the definition \cite[Def. 15.1]{cank} of a relative idempotent completion. 
  In the following we let
   $\bE$ be 
     the full subcategory   of $\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$ with the same objects as $\bD$.  Then $\bD$ is an ideal in $\bE$ and
     the  idempotent completion
  $\bD\to \mathrm{Idem}(\bD)$   is understood relative to $\bE$.

 
Recall the notion of a relative equivalence from \cite[Def. 15.4]{cank}.
\begin{lem}
The inclusion
$\mathrm{Idem}(\bD)\to \mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)$ is a relative equivalence.
\end{lem}
\begin{proof}
By   definition, the objects of  $\mathrm{Idem}(\bD)$  are the objects of $\mathrm{Idem}(\bE)$.
The latter is a full subcategory of $\mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$.
Furthermore, the morphisms of $\mathrm{Idem}(\bD)$ are those morphisms of $\mathrm{Idem}(\bE)$ which belong to the ideal $\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)$ of $\mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$. In other words, we 
  have a pull-back diagram
$$\xymatrix{\mathrm{Idem}(\bD)\ar[r]\ar[d]&\mathrm{Idem}( \bE) \ar[d]\\ \mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)\ar[r]&\mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)}$$
of $C^{*}$-categories.
The right  vertical morphism  in ${\mathbf{C}^{\ast}\mathbf{Cat}}$ is fully faithful by definition. Since $\bE$ contains the all objects  of the form
$(C,\rho,\tilde p_{Y})=(C,\rho)$  for  free $G$-sets $Y$  and $(C,\rho,\mu)$ in ${\mathbf{C}}^{(G)}_{\mathrm{lf}}(Y_{min})$ conclude that it is essentially surjective.  
\end{proof}

This finishes the proof of Proposition \ref{wrtoihjwogregergwergwreg}.
 \end{proof}

We now apply the cone sequence \eqref{wefqwefqewewfwdqewdewdqede} to the functor 
 $K\bC\cX^{G}_{c,G_{can,min}}$ and obtain a boundary map
 $$\partial^{{\mathtt{Cone}}}:K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))\to \Sigma K\bC\cX^{G}_{c,G_{can,min}}(-)$$
 of functors
 from $G\mathbf{UBC}$ to $\Sp^{\mathrm{}}$.

{\begin{ddd} \label{werigowerferfrwef}We let $G\mathbf{UBC}_{\mathrm{bd}}$ be the full subcategory of $G\mathbf{UBC}$ of bounded $G$-uniform bornological coarse spaces. \end{ddd} }We have a forgetful functor $G\mathbf{UBC}_{\mathrm{bd}}\to G\mathbf{BC}_{\mathrm{bd}}$ which we will always drop from the notation. We can restrict the cone boundary
transformation along the inclusion $G\mathbf{UBC}_{\mathrm{bd}}\to G\mathbf{UBC}$.
  
 Let $X$ be in $G\mathbf{UBC}_{\mathrm{bd}}$. 
 We use Corollary \ref{wopijgowergerfwerfwerfewrf}.\ref{wegiojreogewrfwerffw} in order to see the that the natural transformation defined  below  takes values in the correct target.
 \begin{ddd}
 We define the natural transformation 
 $$\mathrm{Asmbl}^{\Theta}:=\theta \circ \partial^{{\mathtt{Cone}}} :K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))\to  \underline{\Sigma \mathrm{KK}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}$$
 of functors from $G\mathbf{UBC}_{\mathrm{bd}}$ to $\Sp^{\mathrm{}}$.
   \end{ddd}
   
   
   

We consider the functor \begin{equation}\label{}
\tilde \iota:G\mathbf{Orb}\to G\mathbf{UBC}_{\mathrm{bd}}\ , \quad S\mapsto S_{min,max,disc}\ ,
\end{equation}  (compare with \eqref{adfvoijqiorjvoifvafsvsdv})
where $disc$ stands for the discrete uniform structure.  
{Note that $\tilde \iota^{*} K\bC\cX^{G}_{c,G_{can,min}}\simeq K\bC^{G}$ by Definition \ref{qroeigjoqergerqfewewfqewfeqwf}.  
We use this identification in order to interpret the target of the map in Assertion \ref{qwhioqwehfiouqewfqewqewfqf} below.}


A $G$-simplicial complex is a simplicial complex with a simplicial $G$-action.
We assume that if $g$ in $G$ fixes a point in the interior of a simplex, then it fixes the whole simplex pointwise.  This can always be ensured by going over to a barycentric subdivision.
We let $G\mathbf{Simpl}$
denote the category of $G$-simplicial complexes and simplicial equivariant maps.  

Let $G\mathbf{Simpl}^{\mathrm{fin}-\dim}$ 
 be the full subcategory of $G\mathbf{Simpl}$ of finite-dimensional $G$-simplicial complexes. 
We have a natural functor
$$\tilde s:G\mathbf{Simpl}\to G\mathbf{UBC}_{\mathrm{bd}}$$
which sends a $G$-simplicial complex $X$ to the $G$-uniform bornological coarse space
$\tilde s(X)$ 
given by $X$ with the coarse and the uniform structures induced by the spherical path metric, and with the maximal bornology. We have a commutative diagram 
of canonical functors   
$$\xymatrix{G\Set\ar[rr]^{  \iota}\ar[dr]&&G\mathbf{UBC}_{\mathrm{bd}}\ar[dr]&&\\&G\mathbf{Simpl}\ar[ur]^{\tilde s}\ar[rr]^(0.3){\tilde t}&&G\Top\\&&G\mathbf{Simpl}^{\textrm{fin-dim}}\ar[lu]\ar[uu]^(0.3){s}\ar[ru]^{t}&&}$$
where the left down-right arrow interprets a $G$-set as a zero-dimensional $G$-simplicial set, and the other un-named arrow sends a uniform bornological space to the underlying $G$-topological space.



\begin{prop}\label{weqfiuhqiuwhefiqwefewewfqfwefqwef}
\mbox{}
\begin{enumerate}
\item \label{qwhioqwehfiouqewfqewqewfqf} The transformation $ \iota^{*}\partial^{{\mathtt{Cone}}}: \iota^{*}K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))\to \Sigma K\bC^{G}(-)$ of functors from $G\Set$ to $\Sp^{\mathrm{}}$ is an equivalence.
\item \label{qfeiohqoifweqewfqf}  We have an  equivalence \begin{equation}\label{adsvkanjovasdvassdvavads}
s^{*}K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))\simeq  t^{*}\Sigma K\bC^{G}(-)
\end{equation}  of functors from $G\mathbf{Simpl}^{\textrm{fin-dim}}$ to $\Sp^{\mathrm{}}$.
\item \label{fopepofkopqwkfpoqwefkqwefqwefqe}  We have a commutative square of natural transformations
  \begin{equation}\label{qerffqwefwfwefewqfwf} \xymatrix{ {t^{*}}\Sigma K\bC^{G}(-)\ar[rr]^{\Sigma \mathrm{Asmbl}^{h}}\ar@{-}[d]_{\simeq}^{\eqref{adsvkanjovasdvassdvavads}}&&\underline{\Sigma K\bC^{G}(*)}\ar@{-}[d]_{\eqref{wrtoihjwogregergwergwreg}}^{\simeq}\\   {s^{*}}K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))\ar[rr]^{\mathrm{Asmbl}^{\Theta}}&&  \underline{\Sigma \mathrm{KK}(\C,  \tilde {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}} 
\end{equation}
between functors from $G\mathbf{Simpl}^{\mathrm{fin}-\dim}$ to $\Sp$.

\end{enumerate}
\end{prop}
\begin{proof}
The Assertion  \ref{weqfiuhqiuwhefiqwefewewfqfwefqwef}.\ref{qwhioqwehfiouqewfqewqewfqf}
is shown in \cite[Prop. 9.35]{equicoarse}. 

 
 
We now show Assertion  \ref{weqfiuhqiuwhefiqwefewewfqfwefqwef}.\ref{qfeiohqoifweqewfqf}.

\begin{rem}Note that $K\bC^{G}(-)$ in the statement is  the restriction  to $G\mathbf{Simpl}^{\textrm{fin-dim}}$ of the evaluation of an equivariant homology   theory defined on all of $G\Top$ by \eqref{adfiuhuihavfvasdcacsadcasca}.     The 
other functor $K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))$ is defined on $G\mathbf{UBC}$. By restriction of both functors  
to $G\mathbf{Simpl}$ we can consider them on the same domain. Assertion  \ref{weqfiuhqiuwhefiqwefewewfqfwefqwef}.\ref{qwhioqwehfiouqewfqewqewfqf} in  then provides equivalence between the further restriction of both functors  to the orbit category.
The functor  $K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))$ is homotopy invariant and excisive for closed decompositions,
  but it is not the restriction of an equivariant homology theory. Therefore  
  we can not appeal to  the classification of equivariant homologies in order to extend this transformation to
  more general spaces. 


One could use Corollary  \ref{wegtopwgwgregergw} applied to $K\bC^{G}$ and $K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))$ in place of $E$ and $F$ and for the family $\cF:=\mathbf{All}
$  in order to obtain an equivalence   on $G$-finite $G$-simplicial complexes.


Our argument below uses that Assertion \ref{qwhioqwehfiouqewfqewqewfqf} provides an equivalence on $G\Set$ (as opposed to $G\mathbf{Orb}$) and yields a more general result:
 A natural equivalence between the restrictions of two homotopy invariant and excisive functors  to $G\Set$  canonically extends to an equivalence between these functors at least on $G\mathbf{Simpl}^{\textrm{fin-dim}}$.\hspace*{\fill}$\qed$ \end{rem}
  
 
We will construct the desired equivalence by   induction with respect to the dimension. 
We let $G\mathbf{Simpl}_{\le n}$ be the full subcategory of $G$-simplicial complexes of dimension $\le n$.
We let $s_{n}$ and $t_{n}$ denote the restrictions of $s$ and $t$ to $G\mathbf{Simpl}_{\le n}$.

The case of zero-dimensional simplicial complexes is  done by Assertion \ref{qwhioqwehfiouqewfqewqewfqf}. 


 We assume  {now} that we have constructed an equivalence
  \begin{equation}\label{weergwrgkpdsdsdsdogergwe}
s_{n-1}^{*}K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))\simeq  t_{n-1}^{*}\Sigma K\bC^{G}(-)
\end{equation}
for $n\ge 1$.
The induction step exploits the fact that
$s_{n}(X)$ in $G\mathbf{Simpl}_{\le n}$  has a canonical decomposition 
$(Y,Z)$, where $Z$ is the disjoint union of $2/3$-scaled $n$-simplices,
and $Y$ is  the complement of the disjoint union of the interiors of the $1/3$-scaled $n$-simplices 
 (see the pictures in \cite[p. 80]{buen}).
 {We equip the subspaces $Y$ and $Z$ with the uniform bornological coarse structures induced from $s_{n}(X)$.}


Since both functors $s^{*}_{n}K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))$ and
$ t_{n}^{*}\Sigma K\bC^{G}(-)$ are excisive for such decompositions
we get push-out squares
\begin{equation}
\xymatrix{K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(Y\cap Z)) \ar[r]\ar[d]&K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(Z)) \ar@{..>}[d]\\K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(Y)) \ar@{..>}[r]&K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}({s_{n}(X)}))}
\end{equation} and
\begin{equation}
\xymatrix{\Sigma K\bC^{G}  (Y\cap Z) \ar[r]\ar[d]&\Sigma K\bC^{G}( Z)\ar@{..>}[d]\\\Sigma K\bC^{G} ( Y)\ar@{..>}[r]&\Sigma K\bC^{G} ( {t_{n}(X)})}
\end{equation}
We now use that both functors are homotopy invariant.
The projection of $Z$ to the $G$-set $Z_{0}$ of barycenters is a homotopy equivalence in $G\Top$ and $G\mathbf{UBC}_{\mathrm{bd}}$. Similarly, there is a projection of $Y$ to the $(n-1)$-skeleton  $X_{n-1}$ of $X$ and a projection of $Y\cap Z$ to a disjoint union $(Y\cap Z)_{n-1}$ of boundaries of the $n$-simplices   which are both homotopy equivalences in  $G\Top$ and $G\mathbf{UBC}_{\mathrm{bd}}$.  These projections identify the bold parts of the push-out squares above canonically with the respective bold parts push-put squares below:
\begin{equation}\label{ewffdfadsfadfasdfasdf}
\xymatrix{K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}({s_{n-1}}(Y\cap Z)_{n-1}) \ar[r]\ar[d]&K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}({s_{0}}(Z_{0})))\ar@{..>}[d]\\K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}({s_{n-1}(X_{n-1})}))\ar@{..>}[r]&K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}({s_{n}}(X)))}
\end{equation} and   
\begin{equation}\label{ewffdfadsfadfasdfasdf1}
\xymatrix{\Sigma K\bC^{G}  ({t_{n-1}}(Y\cap Z)_{n-1}) \ar[r]\ar[d]&\Sigma K\bC^{G}( {t_{0}}(Z_{0}))\ar@{..>}[d]\\\Sigma  K\bC^{G} ( {t_{n-1}}(Y_{n-1}))\ar@{..>}[r]&\Sigma K\bC^{G} ( {t_{n}}(X))}
\end{equation}
The induction hypothesis now provides an equivalence between the bold parts of 
\eqref{ewffdfadsfadfasdfasdf} and \eqref{ewffdfadsfadfasdfasdf1}. This equivalence then provides the desired equivalence of push-outs
$$K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}({s_{n}(}X)))\simeq \Sigma  K\bC^{G} ( {t_{n}(}X))\, .$$
The whole construction is functorial in $X$. To see this interpret  the symbols $X,Y,Z$ as placeholders for entries of diagram valued functors. 


 Assertion 
 \ref{weqfiuhqiuwhefiqwefewewfqfwefqwef}.\ref{fopepofkopqwkfpoqwefkqwefqwefqe}.
  becomes obvious if we 
  expand the square  \eqref{qerffqwefwfwefewqfwf} as follows 
  \begin{equation}
  \xymatrix{ {t^{*}}\Sigma K\bC^{G}(-)\ar[r]^{\mathrm{Asmbl}^{h}} \ar[r]&\Sigma K\bC^{G}(*) \ar[dr]_{\simeq }^{ \theta_{*},\eqref{qwefweofjwpoefeqfeqwfqwe}} \\ {s^{*}}K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(-))\ar@{-}[u]^{\simeq}_{\eqref{adsvkanjovasdvassdvavads}} \ar@/_2cm/[rr]^{\mathrm{Asmbl}^{\Theta} }\ar[r] &K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(*))\ar[r]_{     \mathrm{Asmbl}^{\Theta}_{*}}^{  }  \ar[u]^{\simeq}_{\partial^{{\mathtt{Cone}}},\eqref{adsvkanjovasdvassdvavads}}& \Sigma \mathrm{KK}(\C,  \tilde {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)} 
\end{equation}
 The right horizontal maps in the square are induced by the natural transformation $(-)\to \underline{{\mathtt{const}}_{*}}$, and the upper-left square commutes by the naturality    statement  in Assertion   \ref{weqfiuhqiuwhefiqwefewewfqfwefqwef}.\ref{qfeiohqoifweqewfqf}.
 The upper right triangle commutes by the definition of $\mathrm{Asmbl}_{*}^{\Theta}$, and finally the lower triangle commutes by the naturality of $\mathrm{Asmbl}^{\Theta}$.
 \end{proof}




\section{\texorpdfstring{$\boldsymbol{C^{*}}$}{Cstar}-categorical model for  the analytic assembly map}


The analytic assembly map $\mathrm{Asmbl}^{\mathrm{an}}$ in Definition \ref{qeriughioergewgergwer9} was obtained using a construction on the level of spectrum-valued $KK$-theory. If we precompose this assembly map with the Paschke transformation 
from Theorem \ref{troigjwrtoijgergrweg}, then we get a functor  whose domain is also expressed through
the coarse $K$-homology functor $K\bC\cX^{G}_{c}$ and therefore in terms of $C^{*}$-categories of controlled objects.
In the present section we construct an assembly map
$\mathrm{Asmbl}^{\Lambda}$ in terms of a natural functor $\Lambda$ between $C^{*}$-categories which models this composition.
We then relate $\mathrm{Asmbl}^{\Lambda}$ with both, $\mathrm{Asmbl}^{\Theta}$ and $\mathrm{Asmbl}^{\mathrm{an}}$.






 
{\begin{ddd}\label{wtrkogerfrfwerf}We let $G\mathbf{UBC}_{\mathrm{pc}}$ be the full subcategory of $G\mathbf{UBC}$ of $G$-uniform bornological coarse spaces  which have the bornology of relative compact subsets and whose underlying $G$-topological space belongs to $G\mathbf{LCH}_{+,\mathrm{pc}}^{\mathrm{prop}}$ introduced in Definition \ref{wrtoigjiowrgerferfw}.\end{ddd}}  
Let $X$ be in $G\mathbf{UBC}_{\mathrm{pc}}$ and choose
$\chi$ in $\mathcal{R}(X)$, see \eqref{afvoijvodvasdvsdvavdv}. 


If $ (C,\rho,\mu)$ is an object in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes G_{can,max})$, then we can consider the homomorphism $\phi:C_{0}(X)\to \End(X)$ defined in \eqref{ewqfpojlqkrmeflkwefqef}. Note that
\begin{equation}\label{ewqfoihioqwhejfioewqfewfeqfew}
p_{\chi}:=\sum_{m\in G}  (\phi(\chi) \phi(m^{*}\chi),m)
\end{equation} 
is a projection 
   on $(C,\rho)$  considered as an object of 
in $\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G$. The sum in \eqref{ewqfoihioqwhejfioewqfewfeqfew} has finitely many non-zero terms. We refer to Proposition \ref{weoigjweogergrrewgwregfewfwwrg} for the necessary verifications related with the following definiton.
\begin{ddd}\label{erwoighfgojrefqwfewfqwefeqwf}
We define a   functor
$$\Lambda_{(X,\chi)}: \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes G_{can,max})\to  \mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$$ in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$ as follows: 
\begin{enumerate}
\item objects: The  functor  $\Lambda_{(X,\chi)}$ sends the object
$(C,  \rho,\mu)$ in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}(X)\otimes G_{can,max})$ to the object
$(C,\rho,p_{\chi})$ in  $\mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$, where $p_{\chi}$ is as in 
 \eqref{ewqfoihioqwhejfioewqfewfeqfew}
 
\item  morphisms: The  functor $\Lambda_{(X,\chi)}$ sends a 
 morphism 
  $A:(C,\rho,\mu)\to (C',\rho',\mu')$ in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}(X)\otimes G_{can,max})$  
 to the morphism \begin{equation}\label{fqwefqoijofqwefqwefqef}
\Lambda_{(X,\chi)}(A):=\sum_{m\in G}   (  \phi'(m^{*}\chi)
  A \phi(\chi),m) 
\end{equation} in $\mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$.
\end{enumerate}
 \end{ddd}


In order to state the naturality of $\Lambda_{(X,\chi)}$ we introduce the category
$G\mathbf{UBC}^{\mathcal{R}}_{\mathrm{pc}}$ given by the Grothendieck construction of the functor $\mathcal{R}$. Its objects are pairs $(X,\chi)$ of an object $X$ in $G\mathbf{UBC}_{\mathrm{pc}}$ and $\chi$ in $\mathcal{R}(X)$, and a morphism $f:(X,\chi)\to (X',\chi')$ in $G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}$ is a morphism $f:X\to X'$ in $G\mathbf{UBC}_{\mathrm{pc}}$ such that $f^{*}\chi'=\chi$.  We have a forgetful functor $G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}\to G\mathbf{UBC}_{\mathrm{pc}}$ 
which we will not write explicitly in formulas.
 



 \begin{prop}\label{weoigjweogergrrewgwregfewfwwrg}\mbox{}
 \begin{enumerate}
 \item\label{ewtgojkropgwerefwefgregergwerg} The functor  $\Lambda_{(X,\chi)}$ is well-defined.
 \item \label{qroijgoijerogwgergergwergwergw}  The family $(\Lambda_{(X,\chi)})_{(X,\chi)\in G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}}$ is a natural transformation
 $$\Lambda: \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(-)\otimes G_{can,max})\to  \underline{\mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)}$$
 of functors from $G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}$ to $\Sp^{\mathrm{}}$.
  \label{ewtgojkropgwergwefwfwefregergwerg1}
  \item\label{wetoiguwoefmw09i09m}
 The  transformation     restricts to a   transformation      \begin{equation}
 \label{grpogkprwefwefwsdfsdfsdffewgref}
\Lambda :  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(-)\otimes G_{can,max})\to \underline{\mathrm{Idem}( {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}
\end{equation}
of functors from $G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}$ to $\Sp^{\mathrm{}}$.
 \end{enumerate}
 \end{prop}
 \begin{proof}
 The structure of this proof is the same as for Proposition \ref{weoigjweogergrrewgwregwrg}.
 We start with the verification of Assertion \ref{weoigjweogergrrewgwregfewfwwrg}.\ref{ewtgojkropgwerefwefgregergwerg} {consisting of the combination of}    the next three lemmas.
 

\begin{lem} 
The sum in \eqref{fqwefqoijofqwefqwefqef} converges in $\tilde \bC_{\mathrm{std}}^{(G)}\rtimes_{r} G$. 
\end{lem}
\begin{proof}
We first observe that   if $A$ is controlled, then the sum in \eqref{fqwefqoijofqwefqwefqef} has finitely many non-zero terms.
  Since general morphisms in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}(X)\otimes G_{can,max})$ can be approximated by controlled morphisms it 
 suffices to show that 
$$\|\sum_{m\in G}   (  \phi'(m^{*}\chi)
  A \phi(\chi),m)\|\le \|A\|$$
for controlled $A$.
  So from now on assume that $A$ is controlled so that $\Lambda_{(X,\chi)}(A)$ is defined by \eqref{fqwefqoijofqwefqwefqef}. 
 In order to  estimate the norm 
  we apply the  defining representation
$\sigma : \bC_{\mathrm{std}}^{(G)}\rtimes_{r} G\to \mathbf{L}^{2}(G,  \bC_{\mathrm{std}}^{(G)})$ \cite[Rem. 10.5]{cank}
and show that $\|\sigma(\Lambda_{(X,\chi)}(A))\|\le \|A\|$.

Let 
 $( \bigoplus_{l\in G} C, (e_{l})_{l\in G})$   and 
$( \bigoplus_{l\in G} C', (e'_{l})_{l\in G})$
represent the orthogonal sums of the constant families  $(C)_{l\in G}$  and $(C')_{l\in G}$.   
For a morphism $(f,g): (C,\rho)\to (C',\rho')$ in $\tilde \bC_{\mathrm{std}}^{(G)}\rtimes_{r} G$ we have 
$$\sigma(f,g)=\sum_{l\in G} e'_{l} (lg)\cdot (f) e^{*}_{lg}:    \bigoplus_{l\in G} C\to \bigoplus_{l\in G} C'   \ ,$$
 see \cite[(10.7)]{cank}.
Using  the rule $k\cdot\phi(\theta)= \phi(k^{-1,*}\theta)$ for all $k$ in $G$ and $\theta$ in $C_{0}(X)$ we calculate 
\begin{eqnarray*}
\sigma(\Lambda_{(X,\chi)}(A))&=& \sum_{g\in G} \sigma(   \phi'(g^{*}\chi)
  A \phi(\chi),g)\\&=& \sum_{g,l\in G}  e'_{l} (lg)   (lg)\cdot (   \phi'(g^{*}\chi)
  A \phi(\chi))  e^{*}_{lg}\\&=&
   \sum_{g,l\in G}  e'_{l}    \phi'(l^{-1,*}\chi) A \phi((lg)^{-1,*}\chi) e^{*}_{lg}\\
   &=&
   \sum_{k,l\in G}  e'_{l}     \phi'(l^{-1,*}\chi) A \phi(k^{-1,*}\chi) e^{*}_{k}\ .
\end{eqnarray*}

 The family
$(  e'_{l}     \phi'(l^{-1,*}\chi) A )_{l\in G}$ is a mutually orthogonal family of morphisms $ C\to \bigoplus_{l\in G} C'$. In order to see that the sum over $l$ exists we must check that 
$$\sup_{F\in G} \|\sum_{l\in F}  (   \phi'(l^{-1,*}\chi) A )^{*}    \phi'(l^{-1,*}\chi) A       \|
$$ is uniformly bounded w.r.t the choice of $F$ \cite[Cor. 5.1]{cank}.
We calculate
  \begin{eqnarray*}
\lefteqn{\|\sum_{l\in F}  (   \phi'(l^{-1,*}\chi) A )^{*}    \phi'(l^{-1,*}\chi) A \| }&&\\ &=&
\|\sum_{l\in F}  A^{*}   \phi'(l^{-1,*}\chi)     \phi'(l^{-1,*}\chi) A\| \\&=&  \|A^{*}  \sum_{l\in F} \phi'(l^{-1,*}\chi)^{2}    A \|\\&\le& \|A  \|^{2}
\end{eqnarray*}
 because of the operator inequality 
 $$0\le \sum_{l\in F} \phi'(l^{-1,*}\chi)^{2}\le \id_{C'}\ .$$
It follows that
$$\sum_{l\in G}e'_{l}     \phi'(l^{-1,*}\chi) A  :C\to \bigoplus_{l\in G} C'$$
exists and is bounded in norm by $\|A\|$.

We now consider the mutually orthogonal family
$(\sum_{l\in G}e'_{l}     \phi'(l^{-1,*}\chi) A \phi(k^{-1,*}\chi) e^{*}_{k})_{k\in G}$ 
of morphisms
$ \bigoplus_{k\in G} C'\to \bigoplus_{l\in G} C'$.
We want to show that the sum over $k$ exists.  
We must check that
$$\|\sum_{k\in F}  (\sum_{l\in G}e'_{l}     \phi'(l^{-1,*}\chi) A \phi(k^{-1,*}\chi) e^{*}_{k})
 (\sum_{m\in G}e'_{m}     \phi'(m^{-1,*}\chi) A \phi(k^{-1,*}\chi) e^{*}_{k})^{*}
   \|$$ is bounded independently of the finite subset $F$ of $G$.
   We calculate
\begin{eqnarray*}
\lefteqn{\|\sum_{k\in F}  (\sum_{l\in G}e'_{l}     \phi'(l^{-1,*}\chi) A \phi(k^{-1,*}\chi) e^{*}_{k})
 (\sum_{m\in G}e'_{m}     \phi'(m^{-1,*}\chi) A \phi(k^{-1,*}\chi) e^{*}_{k})^{*}\|}&&\\&=&
 \|   \sum_{l,m\in G}e'_{l}     \phi'(l^{-1,*}\chi) A \sum_{k\in F}  ( \phi(k^{-1,*}\chi) e^{*}_{k}
 e_{k} \phi(k^{-1,*}\chi))
  A^{*}  \phi'(m^{-1,*}\chi)e^{\prime,*}_{m} \|\\&\le &
  \| \sum_{l\in G}e'_{l}     \phi'(l^{-1,*}\chi) A\|^{2}\| \sum_{k\in F}  ( \phi(k^{-1,*}\chi) e^{*}_{k}
 e_{k} \phi(k^{-1,*}\chi))\|\\&\le&\| \sum_{k\in F}   \phi(k^{-1,*}\chi)^{2}  \|
 \|A\|^{2}\\&\le&\|A\|^{2}
  \end{eqnarray*}
  This finally shows that $\|\sigma(\Lambda_{(X,\chi)}(A))\|\le \|A\|$.
\end{proof}

 


\begin{lem}
The definition of $\Lambda_{(X,\chi)} $ in \eqref{fqwefqoijofqwefqwefqef} is compatible with the composition and involution.
\end{lem}
\begin{proof}
Let $A:(C,\rho,\mu)\to (C',\rho',\mu')$ and $A':(C',\rho',\mu')\to (C'',\rho'',\mu'')$ be two morphisms in  $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}(X)\otimes G_{can,max})$. We assume that both are controlled. We calculate using \eqref{goqiejgoigqfqwfwefqwef}  and  the relation $g\cdot A=A$ for all $g$ in $G$ at the marked equality:
\begin{eqnarray*}
\Lambda_{(X,\chi)}(A')\Lambda_{(X,\chi)}(A)&=&\sum_{m,n\in G}   (  \phi''( m^{*}\chi)  A'\phi'(\chi),m)   (  \phi'(n^{*} \chi)  A \phi(\chi),n)\\&=&
\sum_{m,n\in G}   ( n^{-1} \cdot ( \phi''(m^{*} \chi) A'\phi'(\chi))  \phi'(n^{*} \chi) \ A \phi(\chi),mn)\\&=&
\sum_{m,n\in G}  (   \phi''( (mn)^{*}\chi))   n^{-1}\cdot (A')     \phi'(n^{*} \chi)^{2} A \phi(\chi),mn)\\&\stackrel{!}{=}&
\sum_{k\in G}  (   \phi''( k^{*}\chi)    A'A \phi(\chi),k)\\
&=& \Lambda_{(X,\chi)}(A'A)\,.
\end{eqnarray*}
Furthermore,
\begin{eqnarray*}
\Lambda_{(X,\chi)}(A)^{*}&=&\sum_{m\in G}   (  \phi'(m^{*}\chi) A \phi(\chi),m)^{*}\\
&=&\sum_{m\in G}   ( m \cdot (\phi'(m^{*}\chi)  A \phi(\chi))^{*},m^{-1})\\
&=&\sum_{m\in G}   (  (\phi(\chi)  A    \phi'(m^{-1,*}\chi))^{*},m^{-1})\\
&=& \sum_{m\in G}   (   \phi'(m^{-1,*}\chi) A^{*}    \phi(\chi)  ,m^{-1})\\
&=&
\Lambda_{(X,\chi)}(A^{*})
\end{eqnarray*}
varifying compatibility with the involution.
\end{proof}

Finally recall that $\Lambda_{(X,\chi)}(A)$ must be a morphism from $(C,\rho,p_{\chi})$ to $(C',\rho',p'_{\chi})$ in the category $\mathrm{Idem}(  \bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$. This is ensured by the following lemma:
\begin{lem}\label{iuheiuhiuqehfiuhvuiasvsdvasdvadsv}
$p'_{\chi} \Lambda_{(X,\chi)}(A)=\Lambda_{(X,\chi)}(A)=\Lambda_{(X,\chi)}(A)p_{\chi}$.
\end{lem}
\begin{proof}
Assume again that $A$ is controlled.
 We calculate:
\begin{eqnarray*}
p'_{\chi}\Lambda_{(X,\chi)}(A)&=&\sum_{n,m\in G}   (\phi'(\chi) \phi'(n^{*}\chi),n)  (  \phi'(m^{*}\chi)  A \phi(\chi),m)\\&=&
\sum_{n,m\in G}   (m^{-1}\cdot (\phi'(\chi) \phi'(n^{*}\chi))     \phi'(m^{*}\chi)  A \phi(\chi),nm)\\&=&
 \sum_{n,m\in G}   ( \phi'(m^{*}\chi) \phi'((nm)^{*}\chi)     \phi'(m^{*}\chi)  A \phi(\chi),nm)\\&=&
 \sum_{k,m\in G}   ( \phi'(m^{*}\chi) \phi'(k^{*}\chi)     \phi'(m^{*}\chi)  A \phi(\chi),k)\\&=& \sum_{k\in G}   (  \phi'(k^{*}\chi)      A \phi(\chi),k)\\&=&\Lambda_{(X,\chi)}(A)
\end{eqnarray*}
Similarly, $\Lambda_{(X,\chi)}(A)p_{\chi}=\Lambda_{(X,\chi)}(A)$.
\end{proof}




{This finishes the verification of Assertion \ref{weoigjweogergrrewgwregfewfwwrg}.\ref{ewtgojkropgwerefwefgregergwerg}.}
We continue {with} Assertion  \ref{weoigjweogergrrewgwregfewfwwrg}.\ref{qroijgoijerogwgergergwergwergw}.
Let $f \colon (X,\chi)\to (X',\chi')$ be a morphism in $G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}$ and note $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(f)(C,\rho,\mu)=(C,\rho,f_{*}\mu)$.  We let $f_{*}(\phi):C_{0}(X')\to \End_{\bC}(C)$  be the homomorphism defined with $f_{*}\mu$. Then we have the relation
$$f_{*}\phi(\theta')=\phi(f^{*}\theta')$$
for all $\theta'$ in $C_{0}(X')$. In particular
$(f_{*}\phi)(\chi')=\phi(\chi)$.
 This relation implies that
$p_{\chi}=p_{\chi'}$ and $\Lambda_{(X',\chi')}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(f)(A))=\Lambda_{(X,\chi)}(A)$ (note 
Definition \ref{wtogjoergrwegreggwege}.\ref{oiwejgiowegregwr}).
 These equalities imply the assertion.
 
 
 
 
 We finally verify Assertion  \ref{weoigjweogergrrewgwregfewfwwrg}.\ref{wetoiguwoefmw09i09m}.
 If $A \colon (C,  \rho,\mu)\to (C',\rho',\mu')$ is a morphism in $ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes G_{can,max})$, then    $A\phi(\chi) $ is in ${\mathbf{K}}$ by Lemma \ref{ewiogjoerwgfdsfdgsfg}.    This implies that $\Lambda_{(X,\chi)}(A)$ is a morphism in the ideal
$ {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G$.
\end{proof}

   
We now consider the cone sequence \eqref{wefqwefqewewfwdqewdewdqede} for $E=K\bC\cX^{G}_{c,G_{can,max}}$  whose boundary is a natural transformation
\begin{equation}\label{eqwfojpoqwefqefewqf}
\partial^{{\mathtt{Cone}}}:K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(-))\to \Sigma K\bC\cX^{G}_{c,G_{can,max}}(-)
\end{equation} 
of functors from $G\mathbf{UBC}$ to $\Sp^{\mathrm{}}$. The canonical inclusions
$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes G_{can,min})\to   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes G_{can,min})$
give a further transformation
\begin{equation} \label{eqwfojpoqwefqefewqf1}
\Sigma K\bC\cX^{G}_{c,G_{can,max}}(-) 
  \xrightarrow{\simeq}  \Sigma K^{C^{*}\mathrm{Cat}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(-)\otimes G_{can,min})) \end{equation}
which is actually an equivalence (see the argument for the left vertical equivalence in \eqref{friuhiufhqiwfewfqwefqfqewf} applied to the case $Y=G_{can,min}$). 
The composition of  the  transformations \eqref{eqwfojpoqwefqefewqf} with the equivalence \eqref{eqwfojpoqwefqefewqf1}  will also  be  called the cone boundary transformation 
$$\hat \partial^{{\mathtt{Cone}}}:K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(-))\to   \Sigma K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(-)\otimes G_{can,max}))$$ of functors from $G\mathbf{UBC}$ to $\Sp^{\mathrm{}}$, but we add the $\hat{-}$ in order to distinguish it from \eqref{eqwfojpoqwefqefewqf}.
\begin{ddd}\label{qeoirgjoqwrfewffewfq9}
 We define the natural transformation
\begin{equation}\label{}
\mathrm{Asmbl}^{\Lambda}:=K^{C^{*}\mathrm{Cat}}(\Lambda)\circ \hat \partial^{{\mathtt{Cone}}}:K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(-))\to \underline{ \Sigma \mathrm{KK}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}
\end{equation}  of functors from $G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}$ to $\Sp^{\mathrm{}}$.
 \end{ddd}

   
{If $X$ is in $G\mathbf{UBC}_{\mathrm{pc}}$ (see Definition \ref{wtrkogerfrfwerf}), then it is $G$-bounded, but not necessarily bounded.}
We let $X_{\cB_{max}}$ denote the object of $G\mathbf{UBC}_{\mathrm{bd}}$ ({see Definition \ref{werigowerferfrwef})}  obtained from $X$ by replacing the bornology of $X$ by the maximal bornology.
  
  
 

\begin{prop}\label{tghquifhiweewfqe}
There is a canonical equivalence of functors  \begin{equation}\label{gu98reug98wugergwergwergwwregwerg}
K\bC\cX_{c,G_{can,min}}({\mathcal{O}}^{\infty}((-)_{\cB_{max}}))\simeq K\bC\cX_{c,G_{can,max}}({\mathcal{O}}^{\infty}(-)))
\end{equation}
from $G\mathbf{UBC}_{{\mathrm{pc}}}$ to $\Sp^{\mathrm{}}$.
\end{prop}
\begin{proof}
  



We  employ the notion of continuous equivalence introduced in \cite [Def. 3.21]{desc}.
  Recall the  Definition \ref{oiguoeigqregregwergregwgrg} of a locally finite subset of a $G$-bornological space.
  In the present situation we have a $G$-coarse space $Z$ with two $G$-bornologies.
  We denote the two objects in $G\mathbf{BC}$ by $Z_{0}$ and $Z_{1}$.
  The identity map of $Z$ is a continuous equivalence between $Z_{0}$
 and $Z_{1}$ if the following conditions on every $G$-invariant subset $L$ of $Z$ are equivalent:
 \begin{enumerate} 
 \item $L$ is locally finite in 
 $Z_{0}$ \item  $L$ is locally finite in    $Z_{1}$.
 \end{enumerate}
   In this case we have an obvious equality in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$  \begin{equation}\label{vapojvoidjvadvavadsv}
 \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Z_{0})=   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Z_{0})\ .
\end{equation}  

 
  \begin{lem}\label{wegoijerogregwegrgwgewergw}
  If $X$ is  in $G\mathbf{UBC}_{\mathrm{pc},G\mathrm{bd}}$, then 
 the pairs of bornological coarse spaces
\begin{eqnarray*}
 X\otimes G_{can,max} &\quad  &  X_{\cB_{max}}\otimes G_{can,min} \\
 {\mathcal{O}} (X)\otimes G_{can,max} &\quad  &  {\mathcal{O}} (X_{\cB_{max}})\otimes G_{can,min} 
 \end{eqnarray*}
are continuously equivalent (by the identity map of the underlying sets).
  \end{lem}
\begin{proof}



We consider the second case. The first is similar and simpler.  Let $L$ be a $G$-invariant subset of
$[0,\infty)\times X\otimes G$.  Let $B$ be a bounded subset of $X$ such that $GB=X$.
For $n$ in $\nat$ and   subset $A$ of $X$ we
consider the intersections $L_{n,e}:=L\cap ([0,n]\times X\times \{e\})$ and $L_{n,A}:=L\cap ([0,n]\times A\times G)$.
\begin{enumerate}
\item   $L$ is locally finite in ${\mathcal{O}}(X)\otimes G_{can,max}$ if and only if  $L_{n,A}$ is finite for every $n$ in $\nat$  and bounded subset $A$ of $X$. In particular $L_{n,B}$ is finite.
Hence $L$ is locally finite in ${\mathcal{O}}(X)\otimes G_{can,max}$ if and only    if $L_{n,X}$ consists of finitely many $G$-orbits for every $n$ in $\nat$. 
\item  If $L$ is locally finite in ${\mathcal{O}}(X_{\cB_{\max}})\otimes G_{can,min}$, if and only if 
$L_{n,e}$ is finite for every $n$ in $\nat$. This is the case exactly if $L_{n,X}$ consists of finitely many $G$-orbits. \qedhere
\end{enumerate}
 \end{proof}


 
 Let $Y$ be  any object in $G\mathbf{BC}$ and $X$ be in $G\mathbf{UBC}$.
  Then we have a  diagram in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$
  \begin{equation}\label{}
\xymatrix{& \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(X\otimes Y)\ar[d] \ar[r] &\ar@{=}[d] \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes Y)\ar[r]& \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}^{\infty}(X)\otimes Y)\\
0\ar[r]& \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes Y)\ar[r]& \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes Y)\ar[r]&\frac{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes Y)}{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes Y)}\ar[r]&0}
\end{equation} which is natural in $X$,  
where the lower sequence is exact,  and where the square commutes. 
If we apply $K^{C^{*}\mathrm{Cat}}$ and use the  Definition \ref{qrogijeqoifefewfefewqffe}, then we get the (natural in $X$) commuting diagram
{\tiny  \begin{equation}\label{friuhiufhqiwfewfqwefqfqewf}
\xymatrix{\ar[r]&K\bC\cX^{G}_{c,Y} (X)\ar[d]^{\simeq}  \ar[r] &\ar@{=}[d]K\bC\cX^{G}_{c,Y}({\mathcal{O}}(X)) \ar[r]&K\bC\cX^{G}_{c,Y}({\mathcal{O}}^{\infty}(X)\otimes Y)\ar[r]\ar@{..>}[d]^{\simeq }&\\
\ar[r]&K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes Y))\ar[r]&K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes Y))\ar[r]&K^{C^{*}\mathrm{Cat}}\left(\frac{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{O}}(X)\otimes Y)}{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes Y)}\right)\ar[r]&}
\end{equation}}The lower sequence is a fibre sequence  by the exactness of $K^{C^{*}\mathrm{Cat}}$, and the upper sequence is an instance of the cone sequence \eqref{wefqwefqewewfwdqewdewdqede}.
We now argue that the left vertical morphism is an equivalence. First of all
for every $n$ in $\nat $ the inclusion 
$$  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Z_{n})\simeq   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Z_{n}\subseteq {\mathcal{O}}(X)\otimes Y)
$$ 
is a  unitary equivalence by \cite[Lem.\ 5.6]{coarsek}, where $Z_{n}:=[0,n]\times X\times Y$ has the structures induced from ${\mathcal{O}}(X)\otimes Y$. The inclusion  $X\otimes Y\to Z_{n}$ given by
$(x,y)\mapsto (0,x,y)$ is a coarse equivalence.
Hence the induced map 
$$K\bC\cX^{G}_{c}( X\otimes Y)\to  K\bC\cX^{G}_{c}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Z_{n}))\stackrel{\simeq}{\to}  K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Z_{n}\subseteq {\mathcal{O}}(X)\otimes Y))$$ is an equivalence for every $n$ in $\nat$. 
We now use that by definition
$$ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes Y)\cong \colim_{n\in \nat}
 \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(Z_{n}\subseteq {\mathcal{O}}(X)\otimes Y)
$$ and that $K^{C^{*}\mathrm{Cat}}$ commutes with filtered colimits by    \cite[Thm.\ 12.4]{cank} (asserting that $K^{C^{*}\mathrm{Cat}}$ is finitary). Hence we get an equivalence
$$K\bC\cX^{G}_{c}( X\otimes Y)\stackrel{\simeq}{\to} K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes Y))$$ induced by the canonical inclusion. This is exactly the left vertical arrow in \eqref{friuhiufhqiwfewfqwefqfqewf}.



  
  We now assume that  $X$ is in $G\mathbf{UBC}_{{\mathrm{pc}}}$.
Using two instances of the the diagram  \eqref{friuhiufhqiwfewfqwefqfqewf}, one for $ X$ and $Y=G_{can,max}$ , and one  for $X_{\cB_{\max}}$ and $Y=G_{can,min}$,  and the equalities of $C^{*}$-categories resulting from Lemma \ref{wegoijerogregwegrgwgewergw} and  \eqref{vapojvoidjvadvavadsv} saying that the corresponding lower  fibre sequences of the two diagrams are equivalent
we get the desired equivalence \eqref{gu98reug98wugergwergwergwwregwerg}.
\end{proof}

  
  Let $(X,\chi)$ be in $G\mathbf{UBC}_{{\mathrm{pc}}}^{\mathcal{R}}$ {(see the text before Proposition \ref{weoigjweogergrrewgwregfewfwwrg})}.
    \begin{prop}\label{erqoiheroigjorgiqfwefqwefqewf}
We have a commuting diagram
\begin{equation}\label{qoirfgjqoirqwfeqewfqfwefq}
\xymatrix{K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(X_{\cB_{\max}}))\ar[rr]^-{ \mathrm{Asmbl}^{\Theta}_{X_{\cB_{max}}}}\ar@{-}[d]_{\eqref{gu98reug98wugergwergwergwwregwerg}}^{\simeq}&&  \Sigma \mathrm{KK}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) 
 \ar@{=}[d]   \\  K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(X ))   \ar[rr]^{\mathrm{Asmbl}^{\Lambda}_{(X,\chi)}}&&  \Sigma \mathrm{KK}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) }
\end{equation}
\end{prop}
\begin{proof}
 Recall the construction of the functor $\Theta$ in  Definition \ref{eoigjoeirgrwegreregrgewrg}
 (see also   \eqref{fqoifoifjoeiwfjqowfqwefqwefqwefq}) and of $\Lambda$ in Definition \ref{erwoighfgojrefqwfewfqwefeqwf}. We  get the following morphism of exact sequences of $C^{*}$-categories.
\begin{align}
\label{rfreffewqewfewfewfqwefqwef}\\
{\small 
\mathclap{
\xymatrix{
0\ar[r]& \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}} (X)\otimes G_{can,max}) \ar@{=}@/_2cm/[dd]\ar[d]^{\Lambda_{(X,\chi)}}\ar[r] & \ar@{=}@/_2cm/[dd] \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}} (X)\otimes G_{can,max}) \ar[d]^{\Lambda_{(X,\chi)}}\ar[r]& \frac{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(  {\mathcal{O}} (X)\otimes G_{can,max})}{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}}(X)\otimes G_{can,max})} \ar@{=}@/_2cm/[dd] \ar[d] \ar[r]&0\\0\ar[r]&\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)  \ar[r]&\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)\ar[r]& \frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)} \ar[r]&0\\ 
0\ar[r]& \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}}(X_{\cB_{max}})\otimes G_{can,min}) \ar[u]_{\Theta_{X_{\cB_{max}}}}\ar[r] & \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}(X_{\cB_{max}})\otimes G_{can,min}) \ar[u]_{\Theta_{{\mathcal{O}}(X_{\cB_{\max}})}}\ar[r]& \frac{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}(  {\mathcal{O}} (X_{\cB_{max}})\otimes G_{can,min})}{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}} (X_{\cB_{max}})\otimes G_{can,min})} \ar[u] \ar[r]&0
}
}
}\notag
\end{align}
The right vertical maps are induced from the universal property of quotients. The round equalities are consequences of Lemma \ref{wegoijerogregwegrgwgewergw}
and \eqref{vapojvoidjvadvavadsv}. The right equality is responsible for the left vertical equivalence in 
\eqref{qoirfgjqoirqwfeqewfqfwefq} up to identifications, see the proof of Proposition \ref{tghquifhiweewfqe}. We apply $K^{C^{*}\mathrm{Cat}}$ and consider the segment of the long exact sequences which involve the boundary map. We use the identification given by the 
right vertical equivalences in the two instances of \eqref{friuhiufhqiwfewfqwefqfqewf}
with $X$ and $G_{can,max}$ and $X_{\cB_{max}}$ and $Y=G_{can,min}$ in order to
express the $K$-theory of the quotient categories in terms of coarse $K$-homology.
\begin{align}\label{weiojgoijregoiergwergwergwegrwergwf}
\mathclap{
\xymatrix{K\bC\cX^{G}_{c,G_{can,max}}({\mathcal{O}}^{\infty}(X))\ar@{-}[dd]_{\simeq}\ar[r]^-{\hat \partial^{{\mathtt{Cone}}}}& \ar@{-}[dd]_{\simeq}\Sigma K^{C^{*}\mathrm{Alg}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}} (X)\otimes G_{can,max}))\ar[rd]^-{K^{C^{*}\mathrm{Alg}}(\Lambda_{(X,\chi)})}&\\&& \mathrm{KK}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)  \\
K\bC\cX^{G}_{c,G_{can,min}}({\mathcal{O}}^{\infty}(X_{\cB_{max}}))\ar[r]^{\hat \partial^{{\mathtt{Cone}}}} &\Sigma K^{C^{*}\mathrm{Alg}}(  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}} (X_{\cB_{max}})\otimes G_{can,min}))\ar[ru]_{K^{C^{*}\mathrm{Alg}}(\Theta_{X_{\cB_{max}}})}&}
}
\end{align}
The left square commutes since it is induced by an equality of exact sequences of $C^{*}$-categories. We must  provide the filler of the right triangle.

This filler will be given by a relative unitary equivalence of functors on the level of $C^{*}$-categories which will be induced from the equivalence provided by the following lemma.
\begin{lem}\label{epgjpfewdfewfqwfewfqef}
The following  triangle is filled by a natural  unitary equivalence  
$$\xymatrix{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}} (X)\otimes G_{can,max}) \ar[dr]^{\Lambda_{(X,\chi)}}\ar@{=}[dd]&\\&\mathrm{Idem}(   \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)\\ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}}(X_{\cB_{max}})\otimes G_{can,min}) \ar[ur]_{\Theta_{X_{\cB_{max}}}}&
}$$
\end{lem}
\begin{proof}
We consider an object $(C,\rho,\mu)$ on the common domain of the functors. Recall that 
 $\mu_{g}:=\mu([0,\infty)\times X\times \{g\})$. Then we define the morphism
$$U:=  \sum_{m\in G}(\phi( \chi )  \mu_{m},m^{-1}):   (C,\rho)\to  (C,\rho) $$  in $\bC^{(G)}_{\mathrm{std}}\rtimes_{r} G$. 
   The convergence of this sum is understood in the sense of multipliers (not in norm) using that $C$ is isomorphic to the orthogonal sum of the family of images of projections $(\mu_{g})_{g\in G}$, see \cite[Sec. 5]{cank}.
   

We calculate
 \begin{eqnarray*}
 U  U^{*}&=& \sum_{m,n\in G}  (\phi( \chi )   \mu_{m},m^{-1}) (\phi( \chi )  \mu_{n},n^{-1})^{*} \\&=& 
\sum_{m,n\in G}  (\phi( \chi )   \mu_{m},m^{-1}) 
( n^{-1} \cdot ( \mu_{n}      \phi( \chi ))  ,n)
 \\&=& \sum_{m,n\in G}  ( n^{-1} \cdot (\phi( \chi )^{2}  \mu_{m}
   \mu_{n}  )  ,m^{-1}n)
 \\&=&  \sum_{m\in G}  ( m^{-1} \cdot (\phi( \chi )^{2}   \mu_{m}
      )  ,e) 
       \\&=&    (  \sum_{m\in G} \phi(m^{*} \chi )^{2}   \mu_{e}
      )  ,e)
       \\
      &=&(\mu_{e},e)\\&=&
      \tilde p
  \end{eqnarray*}
  where $\tilde p$ is as in \eqref{qewfeqwfpokpoqwefqwefqewf}.
  We further calculate
 \begin{eqnarray*}
U^{*}U&=&\sum_{m,n\in G} (   \phi( \chi )   \mu_{n},n^{-1})^{*}    (\phi( \chi )   \mu_{m},m^{-1}) \\&=&
\sum_{m,n\in G}  ( n^{-1} \cdot ( \mu_{n}      \phi( \chi ))  ,n)  (\phi( \chi )   \mu_{m},m^{-1}) \\&=&\sum_{m,n\in G}  ( mn^{-1} \cdot ( \mu_{n}       \phi( \chi ))\phi( \chi )   \mu_{m},nm^{-1}) \\&=&\sum_{m,n\in G}  ( \mu_{m}     \phi((nm^{-1})^{*} \chi )\phi( \chi )   ,nm^{-1}) \\&=&\sum_{m,k\in G}  ( \mu_{m}     \phi(k^{*} \chi )\phi( \chi )    ,k) \\&=&
\sum_{k\in G}  (      \phi(k^{*} \chi )\phi( \chi )    ,k) \\&=& p_{\chi}
\end{eqnarray*}
where $p_{\chi}$ is as in \eqref{ewqfoihioqwhejfioewqfewfeqfew}.
We conclude that
$$U  p_{\chi} =UU^{*}U=\tilde p U\, .$$
We therefore have a unitary  isomorphism in $\mathrm{Idem}(\bC^{(G)}_{\mathrm{std}}\rtimes_{r}G)$.
$$  ( C,\tilde \rho,  p_{\chi })\stackrel{U}{\to}   ( C,  \rho,  \tilde p)$$ as desired.
In order to verify that $U$ implements a natural transformation we must check the
compatibility with morphisms. 
 Let $A:(C,\rho,\mu)\to (C',\rho',\mu')$ be a morphism in the domain of the functors. 
 We let  $U'$ be defined as above for $(C',\rho',\mu')$.
We assume that $A$ is controlled and calculate
\begin{eqnarray*}
U' \Lambda_{(X,\chi)}(A) &=&\sum_{m,g\in G}
   (\phi'(\chi) \mu'_{m},m^{-1}) ( \phi'(g^{*}\chi)A\phi(\chi),g)\\ &=&
\sum_{m,g\in G}
(g^{-1}\cdot(  \phi'(\chi) \mu'_{m} )  \phi'(g^{*}\chi)A\phi(\chi),m^{-1}g)\\&=&
\sum_{m,g\in G}
(   \phi'(g^{*}\chi)^{2} \mu'_{g^{-1}m}  A\phi(\chi),m^{-1}g)\\&=&
\sum_{k,g\in G}
(  \phi'(g^{*}\chi)^{2}  \mu'_{k^{-1}}   A\phi(\chi),k)\\&=&
\sum_{k\in G}
(  \mu'_{k^{-1}}   A\phi(\chi),k)
\end{eqnarray*}

and (recall the definition \eqref{ewqifhwqeifewfewfwefwefqwef} of $\Theta(A)$)
\begin{eqnarray*}
 \Theta_{X}(A) U& =&\sum_{m,g\in G} ( \mu'_{g^{-1}}A\mu_{e},g) (\phi(\chi) \mu_{m},m^{-1})\\&=&\sum_{m,g\in G} ( m\cdot(  \mu'_{g^{-1}}A\mu_{e} ) \phi(\chi) \mu_{m},gm^{-1})\\
 &=& \sum_{m,g\in G} ( \mu'_{mg^{-1}} A \mu_{m} \phi(\chi) ,gm^{-1})\\&=&
  \sum_{m,k\in G} ( \mu'_{k^{-1}}A\phi(\chi), k)\ .
\end{eqnarray*}
We conclude that $\Theta_{X}(A)U=U'\Lambda_{(X,\chi)}(A)$.\end{proof}

The unitary equivalence implements a relative unitary equivalence filling $$\xymatrix{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq {\mathcal{O}} (X)\otimes G_{can,max}) \ar[dr]^{\Lambda_{(X,\chi)}}\ar@{=}[dd]&\\&\mathrm{Idem}(  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)\\ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq  {\mathcal{O}}(X_{\cB_{max}})\otimes G_{can,min}) \ar[ur]_{\quad \Theta_{X_{\cB_{max}}}}&
}$$

We now use \cite[Lem.\ 15.8]{cank} which provides the desired filler of the right  triangle in \eqref{weiojgoijregoiergwergwergwegrwergwf}.
\end{proof}

\begin{rem}
In Proposition \ref{erqoiheroigjorgiqfwefqwefqewf} we could state a stronger assertion 
saying that there is an equivalence of natural transformations from
$G\mathbf{UBC}_{{\mathrm{pc}}}^{\mathcal{R}}$. The constructions    on the $C^{*}$-category level done in the proof  are sufficiently natural.   But writing out the details would amount to write out large higher coherence diagrams.
Since we do not really need this naturality, we refrain from doing so.   \hspace*{\fill}$\qed$
\end{rem}


We consider  $(X,\chi)$  in $G\mathbf{UBC}_{\mathrm{pc}}^{\mathcal{R}}$.  
 We assume that $X$ is in the image of $\iota^{\mathrm{scl}}$ and choose a scale $\tau$ for $X$ such that $(X,\tau)\in G\mathbf{UBC}^{\mathrm{scl}}$ see Definition \ref{qreoighwoefewqffqewffewfq}.  
 Recall the Paschke  morphism $p_{(X,\tau)}$ from  \eqref{qwoeifjoifjewfqwefwfqewfq}.
 We use  Definition \ref{qriofjqofewfefqeqff} in order to rewrite the domain of $\mathrm{Asmbl}_{(X,\chi)}^{\Lambda}$ introduced in Definition \ref{qeoirgjoqwrfewffewfq9}. \begin{prop} \label{erwogijiojfoiqewfewfqwefqwef}
We have a  commuting square \begin{equation}\label{weiojgoijogergwrgwger}
\xymatrix{  K_{\bC}^{G,\cX}( X )  \ar[rr]^-{\mathrm{Asmbl}^{\Lambda}_{(X,\chi)}}\ar[d]_{p_{(X,\tau)}}&&  \Sigma \mathrm{KK}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)\ar@{=}[d]\\   K_{\bC}^{G,\mathrm{An}}(X)\ar[rr]^-{\mathrm{Asmbl}^{\mathrm{an}}_{X}}&&  \Sigma \mathrm{KK}(\C,   {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) } \end{equation}
\end{prop}
\begin{proof}
Recall  the definitions  \eqref{rgfqfewfqewf}, \eqref{rgfqfewfqewf1}  and  \eqref{qwefoujujfqew09ufewewfewfqfqef} of $\bC_{\tau}(X)$, $\bD_{\tau}(X)$ and $\mathbf{Q}_{\tau}(X)$ from  \eqref{qwefoujujfqew09ufewewfewfqfqef}.
  We consider the following commuting diagram  of exact sequences in $C^{*}\mathbf{Cat}^{\mathrm{nu}}$
\begin{align}
\label{rrwer3423ggergwegegwergwegergewge}\\
{\small 
\mathclap{
\xymatrix{0\ar[r]&\bC_{\tau} (X)  \ar[d]^{i} \ar[r] & \bD_{\tau}  (X)\ \ar[d]^{i} \ar[r]&  \mathbf{Q}_{\tau}(X)  \ar[d]^{\bar i} \ar[r]&0\\
0\ar[r]& \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}} (X)\otimes G_{can,max}) \ar[d]^{\Lambda_{(X,\chi)}}\ar[r] &   \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}} (X)\otimes G_{can,max}) \ar[d]^{\Lambda_{(X,\chi)}}\ar[r]& \frac{  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}} (X)\otimes G_{can,max})}{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}}  (X)\otimes G_{can,max})}    \ar[d]^{\bar \Lambda_{(X,\chi)}} \ar[r]&0\\0\ar[r]&\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)  \ar[r]&\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)\ar[r]& \frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)} \ar[r]&0 }
}
}\notag
\end{align}
where $i$ is the canonical inclusion, and $\bar i$ is induced from the universal property of quotients. We use the right vertical  equivalence  of \eqref{friuhiufhqiwfewfqwefqfqewf} for $X$ and $Y=G_{can,max}$ in order to get the equivalence
$$K_{\bC}^{G,\cX}(X)\simeq K^{C^{*}\mathrm{Cat}}\left(\frac{  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{O}} (X)\otimes G_{can,max})}{ \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}( {\mathcal{Z}}\subseteq{\mathcal{O}}  (X)\otimes G_{can,max})}\right)\, .$$
We now expand the square \eqref{weiojgoijogergwrgwger}
as follows:
\begin{align}\label{wiuhiuwehiuehiugwegergreg}\\
\mathclap{
\xymatrix{  K_{\bC}^{G,\cX}( X )  \ar@/^1.5cm/[rrrr]^-{\mathrm{Asmbl}^{\Lambda}_{(X,\chi)}} \ar[rr]^-{\hat \partial^{{\mathtt{Cone}}}} && \Sigma K^{C^{*}\mathrm{Cat}}( \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}({\mathcal{Z}}\subseteq {\mathcal{O}}(X)\otimes G_{can,max}))\ar[rr]^-{K^{C^{*}\mathrm{Cat}}(\Lambda_{(X,\chi)}) } &&  \Sigma \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)\ar@{=}[d]\\
K^{C^{*}\mathrm{Cat}}(\mathbf{Q}_{\tau}(X))\ar[u]_{\simeq}^{K^{C^{*}\mathrm{Cat}}(\bar i)}\ar[d]_{p_{(X,\tau)}}\ar[rr]^-{K^{C^{*}\mathrm{Cat}}(\bar \Lambda_{(X,\chi)}\circ \bar i)} &&K^{C^{*}\mathrm{Cat}}(\frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)})\ar@{=}[d]\ar[rr]^{\partial}&&\ar@{=}[d] \Sigma \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)\\
K_{\bC}^{G,\mathrm{An}}(X)\ar@/_1.5cm/[rrrr]^-{\mathrm{Asmbl}^{\mathrm{an}}_{X}}\ar[rr]^-{\mathrm{ctc}\circ \epsilon^{*}\circ (-\rtimes G)} &&K^{C^{*}\mathrm{Cat}}(\frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)})\ar[rr]^{\partial} &&  \Sigma \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) }
}\notag
\end{align}
The  commutativity of the upper triangle reflects the definition of $\mathrm{Asmbl}_{(X,\chi}^{\Lambda}$ in Definition \ref{qeoirgjoqwrfewffewfq9}. The filler of the middle hexagon is obtained from the naturality of boundary operators using a chase in the morphisms of fibre sequences obtained by applying $K^{C^{*}\mathrm{Cat}}$ to  \eqref{rrwer3423ggergwegegwergwegergewge}.
  It has been shown in Lemma \ref{wrtoihgjoergergegwgrgwerg}
that the upper left vertical  morphism $K^{C^{*}\mathrm{Cat}}(\bar i)$ is an equivalence.
The lower triangle reflects the Definition \ref{qeriughioergewgergwer9}  of $\mathrm{Asmbl}^{\mathrm{an}}_{X}$ where also the notation appearing on the lower left horizontal arrow is explained.  

So in order to produce a filler of the square
\eqref{weiojgoijogergwrgwger} we must provide a filler of the lower left square in \eqref{wiuhiuwehiuehiugwegergreg}. This is the assertion of the following lemma.

\begin{lem}\label{qoirfjqofqwefefqewfq}
We have a commuting square 
$$\xymatrix{ K^{C^{*}\mathrm{Cat}}(\mathbf{Q}_{\tau}(X)) \ar[d]_{p_{(X,\tau)}}\ar[rrr]^-{K^{C^{*}\mathrm{Cat}}(\bar \Lambda_{(X,\chi)} \circ \bar i)} &&&K^{C^{*}\mathrm{Cat}}(\frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)})\ar@{=}[d] \\
K_{\bC}^{G,\mathrm{An}}(X) \ar[rrr]^-{\mathrm{ctc}\circ \epsilon^{*}\circ (-\rtimes G)
}&&&K^{C^{*}\mathrm{Cat}}(\frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)})  }$$
\end{lem}
\begin{proof}


We start with  the following  diagram: 


\begin{equation}\label{erfgeoifjoiewfqwfewfq}
\xymatrix{ \mathrm{KK}^{G}(C_0(X),C_{0}(X)\otimes \mathbf{Q}_{\tau}(X))\ar[r]^-{ \mu_{(X,\tau)},\eqref{oreihoijfvoisfdvervfdsvdfvsvv} }  \ar[d]^{-\rtimes G}&\mathrm{KK}^{G}(C_0(X), \mathbf{Q}^{(G) }_{\mathrm{std}})  \ar[d]^{-\rtimes G}\\ \mathrm{KK}(C_{0}(X)\rtimes G , (C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X)) \rtimes G)\ar[r]^-{ \mu_{(X,\tau)} \rtimes G}\ar[d]^{\epsilon^{*} }
&\mathrm{KK}(C_{0}(X)\rtimes G , \mathbf{Q}^{(G) }_{\mathrm{std}} \rtimes G)\ar[d]^{\epsilon^{*}}\\\mathrm{KK}(\C ,(C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X)) \rtimes G) \ar[r]^{\mu_{(X,\tau)}\rtimes G}
&\mathrm{KK}(\C , \mathbf{Q}^{(G) }_{\mathrm{std}}  \rtimes G) 
} 
\end{equation}
where $\epsilon^{*}$ is given by pre-composition in $\mathrm{KK}$  with the morphism described in \eqref{qewfoihiew9ufj9qweofqewfewfq}. 
The first square commutes since $-\rtimes G$ is a functor.
The second square commutes since $\mathrm{KK}$ is a bifunctor.

The  next diagram extends \eqref{erfgeoifjoiewfqwfewfq} to the left:
\begin{align}\label{qrqoijfqoifewefqwefwef}\\
\mathclap{
\xymatrix{ 
\Hom_{\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} )}( C_0(X),C_0(X))\times   \mathrm{KK}( \C,  \mathbf{Q}_{\tau}(X))   \ar[r]^-{\hatotimes }\ar[d]^{(-\rtimes G)\times \id}&\mathrm{KK}^{G}(C_0(X),C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X))   \ar[d]^{-\rtimes G} \\ 
\Hom_{ C^{*}\mathbf{Alg}^{\mathrm{nu}}}( C_0(X)\rtimes G,C_0(X)\rtimes G) \otimes   \mathrm{KK}( \C,  \mathbf{Q}_{\tau}(X))   \ar[r]^-{\hatotimes }\ar[d]^{\epsilon^{*}\times \id}&\mathrm{KK}(C_{0}(X)\rtimes G , (C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X)) \rtimes G) \ar[d]^{\epsilon^{*}}\\
\Hom_{ C^{*}\mathbf{Alg}^{\mathrm{nu}} }( \C,C_0(X)\rtimes G) \otimes   \mathrm{KK}( \C,  \mathbf{Q}_{\tau}(X))      \ar[r]^-{\hatotimes }&\mathrm{KK}(\C ,(C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X)) \rtimes G) 
 } 
 }\notag
\end{align}
The second square commutes since $\hatotimes $ in \eqref{saCWDQKOIOJ1} is a bifunctor.
The argument for the commutativity of the first square is the same as for the third square in 
\eqref{efqefeqfqefefqewf}.
We finally specialize \eqref{qrqoijfqoifewefqwefwef} it $\id_{C_{0}(X)}$  in $\Hom_{\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}} )}( C_0(X),C_0(X))$ and get 
\begin{equation}\label{qrqoijfqoifewefqwefwerf}
\xymatrix{ 
    \mathrm{KK}( \C,  \mathbf{Q}_{\tau}(X))   \ar[rr]^-{\id_{C_{0}(X)\hatotimes }}\ar@{=}[d] &&\mathrm{KK}^{G}(C_0(X),C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X))   \ar[d]^{-\rtimes G} \\ 
   \mathrm{KK}( \C,  \mathbf{Q}_{\tau}(X))   \ar[rr]^-{\id_{C_{0}(X)}\rtimes G\hatotimes }\ar@{=}[d]  &&\mathrm{KK}(C_{0}(X)\rtimes G , (C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X)) \rtimes G) \ar[d]^{\epsilon^{*}}\\    
   \mathrm{KK}( \C,  \mathbf{Q}_{\tau}(X))      \ar[rr]^-{ \epsilon\hatotimes \id_{ \mathbf{Q}_{\tau}(X)}\rtimes G}&&\mathrm{KK}(\C ,(C_{0}(X)\otimes  \mathbf{Q}_{\tau}(X)) \rtimes G)
 }
\end{equation}
The horizontal composition of \eqref{qrqoijfqoifewefqwefwerf} and \eqref{erfgeoifjoiewfqwfewfq} and using Definition \ref{nlkkmlmvfdvsdva} of $p_{(X,\tau)}$
  yields the bold part of the commuting  diagram
  \begin{equation}
\xymatrix{\mathrm{KK}(\C, \mathbf{Q}_{\tau}(X) \ar[r]^-{p_{(X,\tau)} }\ar@{=}[d]&\mathrm{KK}^{G}(C_{0}(X), \mathbf{Q}^{(G) }_{\mathrm{std}})\ar[d]^{\epsilon^{*}\circ (-\rtimes G)}\\  \mathrm{KK}(\C, \mathbf{Q}_{\tau}(X) \ar[r]\ar@{.>}[dr]_-{K^{C^{*}\mathrm{Cat}}(\Gamma_{(X,\tau,\chi)})\quad}\ar[r]&\mathrm{KK}(\C, \mathbf{Q}^{(G) }_{\mathrm{std}}\rtimes G)  \ar[d]^{\mathrm{ctc}}\\& K^{C^{*}\mathrm{Cat}}(\frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}) }
\end{equation}

 

 Unfolding the definitions we see that  
the dotted   morphism is induced by a  functor
\begin{equation}\label{qwojoijqwvfwevqvwwc}
\Gamma_{(X,\tau,\chi)}:\mathbf{Q}_{\tau}(X)\to \frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}
\end{equation}
which has the following description:
\begin{enumerate}
\item objects:  The functor $\Gamma_{(X,\tau,\chi)}$ sends the object $(C,\rho,\mu)$ in $\mathbf{Q}_{\tau}(X)$  to the object $( C,\rho )$ in $ \frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}
$. 
\item morphisms: The functor $\Gamma_{(X,\tau,\chi)}$ sends a  morphism  $[A]:(C,\rho,\mu)\to (C',\rho',\mu')$ in   $\mathbf{Q}_{\tau}(X)$  to the morphism  
$$\sum_{g\in G} ([\phi'(\chi) \phi'(g^{*}\chi) A], g):( C,\rho )\to ( C',\rho )$$ 
in $ \frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}$. Here we use the formula \eqref{398z9823zf983rferwfwefwrffwr} for $p_{\chi}$ which enters the definition of $\epsilon^{*}$. 

\end{enumerate}

In order to show the lemma we must provide an equivalence \begin{equation}\label{adsoijoiajfoiafqasdfsfs}
K^{C^{*}\mathrm{Cat}}(\Gamma_{(X,\tau,\chi)})\simeq K^{C^{*}\mathrm{Cat}}(\bar \Lambda_{(X,\chi)}\circ \bar i)\ ,
\end{equation}
where  
\begin{equation}\label{weqfiuhiuhiu}
\bar \Lambda_{(X,\chi)}\circ \bar i: \mathbf{Q}_{\tau}(X) \to  \frac{\mathrm{Idem}( \bC^{(G)}_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)} 
\end{equation} is given by the right vertical composition in \eqref{rrwer3423ggergwegegwergwegergewge}.
It has the following explicit description derived from Definition \ref{erwoighfgojrefqwfewfqwefeqwf}:
\begin{enumerate}
\item objects:  The functor $ \bar \Lambda_{(X,\chi)}\circ \bar i$  sends the object $(C,\rho,\mu)$ in $\mathbf{Q}_{\tau}(X)$  to the object $   (C,\rho ,p_{\chi})$ in $  \frac{\mathrm{Idem}( \bC^{(G) }_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G) }_{\mathrm{std}}\rtimes_{r} G)}$. 
\item morphisms: The functor $ \bar \Lambda_{(X,\chi)}\circ \bar i$ sends a  morphism  $[A]:(C,\rho,\mu)\to (C',\rho',\mu')$ in   $\mathbf{Q}_{\tau}(X)$  to the morphism  
$$\sum_{g\in G} ([\phi'(\chi) \phi'(g^{*}\chi) A], g):( C,\rho, p_{\chi})\to ( C',\rho ',p'_{\chi})$$ 
in   $\frac{\mathrm{Idem}( \bC^{(G) }_{\mathrm{std}}\rtimes_{r} G)}{\mathrm{Idem}({\mathbf{K}}^{(G) }_{\mathrm{std}}\rtimes_{r} G)}$. 

\end{enumerate}




Recall the notion of a  relative  MvN equivalence of functors from \cite[Def. 15.9]{cank}.
We claim that 
the functors $ \bar \Lambda_{(X,\chi)}\circ \bar i$ and $ \Gamma_{(X,\tau,\chi)}$ are  relatively (actually absolutely since the targets are unital)  MvN  equivalent. The claim implies the equivalence \eqref{adsoijoiajfoiafqasdfsfs}  by \cite[Prop. 15.10]{cank}. 
 
 The MvN equivalence $v: \bar \Lambda_{(X,\chi)}\circ \bar i\to  \Gamma_{(X,\tau,\chi)}$ is given by the family of partial isometries
 $v=(v_{(C,\rho,\mu)})_{(C,\rho,\mu)\in \mathbf{Q}_{\tau}(X)}$, where
 $v_{(C,\rho,\mu)}:(C,\rho,p_{\chi})\to (C,\rho)$
 is the canonical inclusion. Naturality of $v$ is obvious since the formulas for the action of the functors on morphisms coincide. This finishes the proof of Lemma \ref{qoirfjqofqwefefqewfq}.
  \end{proof}  The proof of Proposition \ref{erwogijiojfoiqewfewfqwefqwef} is now complete.
 \end{proof}


{In the following we omit the functor $t$ sending a $G$-simplicial complex to its underlying $G$-topological space {from the notation}, (e.g.\ at the left upper coerner of the square \eqref{jfjoiwefjwqefwfwefqwefwefq} below).}
\begin{prop}
Assume:
\begin{enumerate}
\item $X$ is a $G$-finite $G$-simplicial complex with finite stabilizers.
\item ${\mathbf{K}}$ is compatible with orthogonal sums.
\end{enumerate}
Then we have a commuting square
\begin{equation}\label{jfjoiwefjwqefwfwefqwefwefq}
\xymatrix{\Sigma K\bC^{G}(X)\ar[r]^-{\Sigma \mathrm{Asmbl}_{X}^{h}}\ar@{-}[d]^{\simeq}&\Sigma K\bC^{G}(*)\ar[d]^{\simeq }\\ K^{G,\mathrm{An}}_{\bC}(X)\ar[r]^-{\mathrm{Asmbl}_{X}^{\mathrm{an}}}&\Sigma \mathrm{KK}(\C,  {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G)}
\end{equation}
\end{prop}
\begin{proof}
Note that $X$ is in $G\mathbf{UBC}_{{\mathrm{pc}}}$ {(see Definition \ref{wtrkogerfrfwerf})}. 
 We choose $\chi$ in $\mathcal{R}(X)$. Furthermore, by Proposition  \ref{qwfiuhqwifweewfefffqedqwedew}.\ref{ergoijeogwegregwgergwgwerg}   {the space} $X$  admits a functorial  lift $(X,\tau)$ in $G\mathbf{UBC}^{\mathrm{scl}}$. {We then consider the diagram}
\[\begin{tikzcd}[column sep=large]
	\Sigma K\bC^G(X) \ar[r,"{\Sigma \mathrm{Asmbl}_X^h}"] \ar[d,"{\ref{weqfiuhqiuwhefiqwefewewfqfwefqwef}}","\simeq"'] & \Sigma K\bC^G(*) \ar[d,"{\ref{wrtoihjwogregergwergwreg}}"',"\simeq"] \\
	K\bC\cX^G_{c,G_{can,min}}({\mathcal{O}}^\infty(X_{{\cB_{max}}})) \ar[r,"{\mathrm{Asmbl}^\Theta_X}"] \ar[d,"{\ref{tghquifhiweewfqe}}","\simeq"'] & \Sigma \mathrm{KK}(\C,  \tilde {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) \ar[d,equal] \\
	K\bC\cX^G_{c,G_{can,max}}({\mathcal{O}}^\infty(X)) \ar[r,"{\mathrm{Asmbl}^\Lambda_{(X,\chi)}}"] \ar[d,equal,"{\mathrm{def}}"] & \Sigma \mathrm{KK}(\C,  \tilde {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) \ar[d,equal] \\ 
	K^{G,\cX}_{\bC}(X) \ar[r,"{\mathrm{Asmbl}^\Lambda_{(X,\chi)}}"] \ar[d,"{p_{(X,\tau)}}","\simeq"'] & \Sigma \mathrm{KK}(\C,  \tilde {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) \ar[d,equal] \\ 
	K^{G,\mathrm{An}}_{\bC}(X) \ar[r,"\mathrm{Asmbl}^\mathrm{an}_X"] & \Sigma \mathrm{KK}(\C,  \tilde {\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r}G) \\
\end{tikzcd}\]
{The} lowest left vertical map is an equivalence by an application of our main Theorem~\ref{qreoigjoergegqrgqerqfewf}.\ref{qregiojqwfewfqwfqewf}.
{The statement that each of the above squares commute is proven, from top to bottom, in Proposition~\ref{weqfiuhqiuwhefiqwefewewfqfwefqwef}.\ref{fopepofkopqwkfpoqwefkqwefqwefqe}, Proposition~\ref{erqoiheroigjorgiqfwefqwefqewf}, the definitions, and Proposition~\ref{erwogijiojfoiqewfewfqwefqwef}. This shows the proposition.}
\end{proof}


 

\phantomsection   \label{wtoiguwegwegr3gergreggrgergregwe}
  \begin{proof}[Proof of Theorem \ref{wtoiguwegwergergregwe}]
  We choose a model for $E_{\cF}G^\mathrm{CW}$ which is a $G$-simplicial complex. 
  Then we apply $\pi_{*}$ to the square \eqref{jfjoiwefjwqefwfwefqwefwefq} and form the colimit   of the resulting squares of groups for $X$ running over the $G$-finite subcomplexes of $E_{\cF}G$.
  This yields \eqref{avvoijfoivjoafvasddsvdvasdv}. 
     \end{proof}
 

\begin{rem} In the proof of Theorem \ref{wtoiguwegwergergregwe} 
 we must apply $\pi_{*}$ before taking the colimit of the subcomplexes. The reason is that we have only constructed the boundary of the square  \eqref{jfjoiwefjwqefwfwefqwefwefq}
 naturally in $X$. For the fillers we just have shown existence for every $X$ separately.
 \hspace*{\fill}$\qed$
\end{rem}



\section{Davis--L\"uck functors and the argument of Kranz} \label{thelast}
  
 In this section we first review the argument of Kranz \cite{kranz} for the comparison of the Davis-L\"uck assembly map with the Kasparov assembly map which involves the Meyer-Nest assembly map as an intermediate step. 
{In more detail, Kranz compares the Davis--L\"uck assembly map with the Meyer--Nest assembly map, which is known to coincide with the analytical assembly map. We will review these comparisons below. In fact, Kranz' paper has two separate parts. On the one hand, he shows that the Davis--L\"uck assembly map associated to a functor}
 $$K^{G}\colon \mathrm{KK}_{\sepa}^{G}\to \Fun(G\mathbf{Orb},\Sp)$$ satisfying certain axioms (stated in Assumption \ref{tgioewrgergwerg}) {is equivalent to the Meyer--Nest assembly map. On the other hand, he provides a concrete construction of such a functor $  K^{G}$.}
We will recall this construction in detail with the goal of showing that it only involves
 formal manipulations using the calculus of equivariant $KK$-theory as developed in \cite{KKG}. {If $A$ is in $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Alg}}_{\mathrm{sep}})$, then  in Proposition \ref{weroigjoewrgergergfewrf} we will show}  that the resulting Davis-L\"uck functor 
 is equivalent to the functor $K\bC^{G}$
 used in the main body of the present paper if we set $\bC:=\mathbf{Hilb}^{G}(A)$ and take the ideal ${\mathbf{K}}:=\mathbf{Hilb}^{G}_{c}(A)$ of compact operators.

 In    Proposition \ref{eqriogergreqfqwfefq} we will apply this comparison in order to show that the analytic assembly map $\mathrm{Asmbl}^{\mathrm{an}}_{\mathbf{Fin}}$ is an equivalence for this choice of $(\bC,{\mathbf{K}})$ provided $A$ is compactly induced {in the sense of Definition~\ref{def:CI-and-CC}}. {We finally show in Proposition \ref{weiojgwoerferww} that in this case,  {the map} 
 $\mathrm{Asmbl}^{\mathrm{an}}_{\cF}$ is equivalent to  {the map} $\mu^{Kasp}_{\cF,A}$.  {Therefore,
results on $\mathrm{Asmbl}^{\mathrm{an}}_{\cF}$ are equivalently results on $\mu^{Kasp}_{\cF,A}$.}
 
 
 
 
 
 

We first recall the Meyer--Nest approach to the Baum--Connes assembly map  \cite{MR2193334}. {Given the results of  \cite{KKG} and the present paper, we will  give an almost self-contained treatment, the only exception is  the usage of  \cite[Prop.\ 4.6]{MR2193334} in the proof of  Proposition \ref{weg8w9egwergweg1} below.}  We    interpret the terminology introduced in \cite{MR2193334} in the stable $\infty$-category $\mathrm{KK}_{\sepa}^{G}$ introduced in \cite[Def. 1.8]{KKG} instead of the triangulated homotopy category of $\mathrm{KK}_{\sepa}^{G}$ as considered   by Meyer--Nest.
We call a subcategory of $\mathrm{KK}_{\sepa}^{G}$ localizing\footnote{Usually, localizing subcategories are stable and cocomplete subcategories of cocomplete stable $\infty$-categories. Since $\mathrm{KK}_{\sepa}^{G}$ is only known to admit countable colimits, we must use this ad-hoc definition.} if it is thick and closed under countable direct sums. In the following we use the restriction,  induction and crossed-product functors   on the level of stable $\infty$-categories as introduced in  \cite[Sec. 1.5]{KKG}. 
\begin{ddd}\label{def:CI-and-CC}\mbox{}
 \begin{enumerate} \item
 We define $\mathcal{CI}$ as the localizing subcategory of $\mathrm{KK}_{\sepa}^{G}$ generated by the objects of the form $\Ind_{H,s}^G(A)$ for  all finite subgroups $H$    of $G$ and objects $A $ in $\mathrm{KK}_{\mathrm{sep}}^{H}$. {The objects of $\mathcal{CI}$ will be called compactly induced.}
 \item We 
 define   $\mathcal{CC}$ as the localizing subcategory of $\mathrm{KK}_{\sepa}^{G}$  given by  all objects $A$ with $\mathrm{Res}_{H,s}^G(A) = 0$ for all finite subgroups $H$ of $G$. \end{enumerate}
\end{ddd}
We note here that $\mathcal{CC}$ is localising because the restriction functors commute with countable  sums  \cite[Lemma 4.3]{KKG}.
The proof of the following proposition is based on  a general adjoint functor theorem applicable in this situation.
\begin{prop} \label{weg8w9egwergweg1}
There exists an adjunction  \begin{equation}\label{wefqwdxcassdcsdca}
\mathrm{incl}:\mathcal{CI}\leftrightarrows  \mathrm{KK}_{\sepa}^{G}:C
\end{equation}
    \end{prop}  
\begin{proof}
For any object $A$ in $\mathrm{KK}_{\sepa}^{G} $  by \cite[Prop.\ 4.6]{MR2193334} there exists an object $\tilde A$ in $\mathcal{CI}$ with a morphism $\tilde A\to A$ (called the Dirac morphism) inducing an equivalence of functors $\mathrm{KK}_{\sepa}^{G}(-,\tilde A)\to \mathrm{KK}_{\sepa}^{G}(-,A)$ from $\mathcal{CI}^{\mathrm{op}}$ to $\Sp$. 
Hence for any $A$ in $\mathrm{KK}_{\sepa}^{G}$ the functor
$\mathrm{KK}_{\sepa}^{G}(-,A)_{|\mathcal{CI}}:\mathcal{CI}^{\mathrm{op}}\to \Sp$ is representable by an object of $\mathcal{CI}$. {This implies the existence of the {right adjoint $C$ to $\mathrm{incl}$}}  {as follows for instance from \cite[Prop.\ 5.1.10]{Land-infinity}.}
\end{proof}


{Let  $\cC$ be a stable $\infty$-category.} Recall that a semi-orthogonal decomposition of $\cC$ is a pair $(\cA,\cB)$ of full stable subcategories such that 
$\map_{\cC}(A,B)\simeq 0$ for all $A$  in $\cA$ and $B$ in $\cB$, and such that 
 for every object $C$ of $\cC$ there exists a fibre sequence
$A\to C\to B$ with $A$ in $\cA$ and $B$ in $\cB$.  {For the sake of  completeness of the presentation, we give the following list of equivalent conditions on a pair $(\cA,\cB)$ of stable subcategories,  {and refer to \cite[Section 7.2.1]{SAG} for more details}:}
\begin{enumerate}
\item\label{ertherhrt1}  The pair $(\cA,\cB)$ is a semi-orthogonal decomposition of $\cC$.
\item The pair $(\cA,\cB)$ is a $t$-structure on $\cC$.
\item\label{ertherhrt3} The inclusion $\cA \to \cC$ has a right adjoint and $\cB$ is the right orthogonal complement of $\cA$.
\item The inclusion $\cB \to \cC$ has a left adjoint and $\cA$ is the left orthogonal complement of $\cB$.
\end{enumerate}

\begin{prop}
\label{weg8w9egwergweg}
The pair $(\mathcal{CI},\mathcal{CC})$ is a semi-orthogonal decomposition of $\mathrm{KK}_{\sepa}^{G}$.
\end{prop}
\begin{proof}
For every subgroup $H$ of $G$ we have an adjunction 
$$\Ind_{H,s}^{G}:\mathrm{KK}_{\mathrm{sep}}^{H} \leftrightarrows  \mathrm{KK}_{\sepa}^{G}: \mathrm{Res}^{G}_{H,s}$$
 which  can be obtained from 
\cite[Thm. 1.22.1]{KKG} by restriction to the separable subcategories.
It is an 
  immediate consequence of the existence of  these adjunctions
  that $\mathrm{KK}_{\sepa}^{G}(A,B)\simeq 0$ for all $A$ in $ {\mathcal{CI}}$ and $B$ in $\mathcal{CC}$. We get in fact the following stronger assertion that $\mathcal{CC}$ consists precisely of the objects 
  $B$ of $\mathrm{KK}_{\sepa}^{G}$ with $\mathrm{KK}_{\sepa}^{G}(A,B)  {\simeq 0}$ for all $A$ in $\mathcal{CI}$, {i.e.\ that $\mathcal{CC}$ is the right orthogonal complement to $\mathcal{CI}$.} 
  
{In view of Proposition \ref{weg8w9egwergweg1}, the following is precisely a specialization of the  argument that Condition \ref{ertherhrt3} above 
 implies Condition \ref{ertherhrt1}.}
We must show  that  for any object $A$ of $\mathrm{KK}_{\sepa}^{G}$, there is a   fibre sequence \begin{equation}\label{erwgwegergewrwferwerc}
 C(A) \longrightarrow A \longrightarrow N(A)\ 
\end{equation}
with $C(A)$ in $\mathcal{CI}$ and $N(A)$ in $\mathcal{CC}$.
By Proposition \ref{weg8w9egwergweg1} we have a fibre sequence of functors
$  C\to \id_{\mathrm{KK}_{\sepa}^{G}}\to N$, where $N:\mathrm{KK}_{\sepa}^{G}\to \mathrm{KK}_{\sepa}^{G}$ is defined as the 
cofibre of the counit of the adjunction {in} \eqref{wefqwdxcassdcsdca}. It suffices to show that $N$ takes values in $\mathcal{CC}$. Let $A$ be in $\mathrm{KK}_{\sepa}^{G}$. Then
for every $B$ in $\mathcal{CI}$ we have
$\mathrm{KK}_{\sepa}^{G}(B,N(A))\simeq \mathrm{cofib}(\mathrm{KK}_{\sepa}^{G}(B,C(A))\to \mathrm{KK}_{\sepa}^{G}(B,A))$.
But $\mathrm{KK}_{\sepa}^{G}(B,C(A))\to \mathrm{KK}_{\sepa}^{G}(B,A)$
 is an equivalence by the construction of $C$
 so that $\mathrm{KK}_{\sepa}^{G}(B,N(A))\simeq 0$. {Since $\mathcal{CC}$ is precisely the right-orthogonal complement of $\mathcal{CI}$} 
   this implies  that $N(A)$  belongs to $\mathcal{CC}$.
 \end{proof}
 

Let $A$ be in $\mathrm{KK}_{\sepa}^{G}$.
\begin{ddd}\label{igowergwergwegwerg}
The Meyer--Nest assembly map for $G$ 
is the map $$\mu^{MN}_{\ast}:\mathrm{KK}_{\sepa}(\C,C(A)\rtimes_r G)\to  \mathrm{KK}_{\sepa}(\C, A\rtimes_r G)$$
induced by $C(A)\to A$ in $\mathrm{KK}_{\sepa}^{G}$.
\end{ddd}

The following theorem is an immediate consequence of \cite[Prop. 5.2]{MR2193334}
which yields the comparison of the Meyer--Nest assembly map and the Kasparov's assembly map. 
\begin{theorem}\label{wrigoghgegrtgeg}
 There is a commutative   square
 $$\xymatrix{  {RK^{G,\mathrm{an}}_{C(A)} (E_{\mathbf{Fin}}G^{\mathrm{CW}})}\ar[r]^{\cong}\ar[d]_{\cong }^{\mu^{Kasp}_{C(A)}}&{RK^{G,\mathrm{an}}_{A} (E_{\mathbf{Fin}}G^{\mathrm{CW}})}\ar[d]^{\mu^{Kasp}_{A}}\\ \mathrm{KK}_{\mathrm{sep}}(\C, C(A)\rtimes_{r}G) \ar[r]^{\mu^{MN}}&\mathrm{KK}_{\mathrm{sep}}(\C, A\rtimes_{r}G)}$$
where the vertical maps are  instances of Kasparov's assembly map of {Definition \ref{wergoijowergerrrfrfrfrfeggwgw}}, and the horizontal maps are induced by the morphism $C(A)\to A$. 
 \end{theorem}
 \begin{proof} 
 {First we note that the square commutes by the naturality of the Kasparov assembly map with respect to the coefficients.} 
{Using Definition \ref{rigrgbjoirjvorgvfbdfgbfgbdfgbdfbg} the upper horizontal map is equivalent to the map 
$$\colim_{W\subseteq E_{\mathbf{Fin}}G^{\mathrm{CW}}}\mathrm{KK}^{G}_{\mathrm{sep}}(C_{0}(W), C(A)) \to  \colim_{W\subseteq E_{\mathbf{Fin}}G^{\mathrm{CW}}}\mathrm{KK}^{G}_{\mathrm{sep}}(C_{0}(W), A)\, ,$$
where the colimits run  over the $G$-finite sub-complexes of $E_{\mathbf{Fin}}G^{\mathrm{CW}}$.
 It is} an  equivalence by the definition of $C(A)\to A$, since $C_{0}(W)$ belongs to  $ \mathcal{CI}$ for every $W$ appearing in the colimit. 
 
 The verification of the fact that $\mu^{Kasp}_{C(A)}$ is an equivalence is more complicated. {The reference \cite{MR2193334}
  employs the work of \cite{OO}
(isomorphism of the induction map) and
\cite[Prop. 2.3]{MR1836047} (compatibility of induction with the Kasparov assembly map).} 
{Using the results of the present paper, Corollary \ref{weiojgwoerferww} gives an independent proof of this fact  {in the case of discrete groups. Note that \cite{MR2193334} considers the more general case of locally compact groups.
}}
\end{proof}

We now consider a functor 
 $$K^{G}:\mathrm{KK}_{\sepa}^{G}\to  \Fun(G\mathbf{Orb},\Sp)\, , \quad A\mapsto K_{A}^{G}\, .$$ 
 In order to formulate the  properties of $K^{G}$ required for Kranz' argument we consider the functor $$i_{H}^{G}:H\mathbf{Orb}\to G\mathbf{Orb}\, , \quad S\mapsto G\times_{H}S$$ and  let $i_{H,!}^{G}$ denote the left Kan extension functor along $i_{H}^{G}$. 
We assume  $K^{G}$ has the following properties:
\begin{ass}\label{tgioewrgergwerg}\mbox{}
 \begin{enumerate}
 \item\label{wrtkhiowgwergefrefw} $K^{G}$ is 
 exact.
 \item {$K^{G}$  preserves  countable  coproducts.}
\item  \label{rgegfgsdfg} For every $A$ in $\mathrm{KK}_{\sepa}^{G}$ and $H$ in $G\mathbf{Orb}$ we have
 an equivalence \begin{equation}\label{fvdsfvsavsdcasdcasdc}
K^{G}_{A}(G/H)\simeq \mathrm{KK}_{\sepa}(\C,(\mathrm{Res}^{G}_{H,s}(A)\rtimes_{r}H)_{s})\, .
\end{equation}  \item \label{giowjwgoegwergwergr} For any  subgroup $H$
  of $G$ we have a commutative square
\begin{equation}\label{cdscasdc}
\xymatrix{\mathrm{KK}_{\mathrm{sep}}^{H}\ar[r]^-{K^{H}}\ar[d]^{\Ind^{G}_{H,s}}&\Fun(H\mathbf{Orb},\Sp)\ar[d]^{i^{G}_{H,!}}\\\mathrm{KK}_{\sepa}^{G}\ar[r]^-{K^{G}}&\Fun(G\mathbf{Orb},\Sp) }
\end{equation}
\end{enumerate}
  \end{ass}

In view of Definition \ref{weigjorgdg} we consider $K^{G}$ as a functor from $\mathrm{KK}_{\sepa}^{G}$ to the stable $\infty$-category of $\Sp$-valued equivariant homology theories. In particular, if $A$ in $\mathrm{KK}_{\sepa}^{G}$ and $X$ is in $G\Top$ we have a well-defined evaluation $K^{G}_{A}(X)$ in $\Sp$.

The argument of Kranz is then based on the following commutative diagram \begin{equation}\label{wergwerrefwercsdcsdc}
 \xymatrix{
	K^G_{C(A)}(E_{\mathbf{Fin}} G^{\mathrm{CW}}) \ar[rr]^-{\mu^{MN}_{E_{\mathbf{Fin}}G^{\mathrm{CW}}}} \ar[d]^{\mu^{DL}_{C(A)}} && K^G_{A}(E_\mathbf{Fin} G^{\mathrm{CW}}) \ar[d]^{\mu^{DL}_{A}} \\
	K^G_{C(A)}(\ast) \ar[rr]^-{\mu^{MN}_{\ast}} && K^G_{A}(\ast)
 }
\end{equation}
 Here the vertical Davis-Lück assembly maps   are induced by the map $E_{\mathbf{Fin}} G^{\mathrm{CW}}\to \ast$\footnote{{Recall that }$\mu^{DL}_{A}$  is $\mathrm{Asmbl}_{K_{A}^{G},E_{\cF}G^{\mathrm{CW}}}^{h}$ from \eqref{qwefeqwdqwed}.}.
 Moreover, the horizontal Mayer-Nest   assembly maps   are  induced by the map $C(A)\to A$ (see \eqref{erwgwegergewrwferwerc}), where $\mu^{MN}_{\ast}$ is indeed the map from Definition \ref{igowergwergwegwerg}.

\begin{theorem}[Kranz]\label{qerighojwergerfewferf}
We have an equivalence $\mu^{DL}_{A}\simeq \mu_{*}^{MN}$. 
\end{theorem}
\begin{proof}

The square in \eqref{wergwerrefwercsdcsdc}  yields an equivalence of $\mu^{DL}_{A}$ with $\mu^{MN}_{*}$  provided one can show that  $\mu^{DL}_{C(A)}$ and $\mu^{MN}_{E_{\mathbf{Fin}}G^{\mathrm{CW}}}
 $ are equivalences. This is the content of the following two  lemmas.
 
 Let $A$ be in $\mathrm{KK}_{\sepa}^{G}$.
 \begin{lem}  The Meyer-Nest assembly map 
 $\mu^{MN}_{E_{\mathbf{Fin}}G^{\mathrm{CW}}}$ is an equivalence.
 \end{lem}
 \begin{proof}
 Since  $K^{G}$ is exact by Assumption \ref{tgioewrgergwerg}.\ref{wrtkhiowgwergefrefw}, 
 using    \eqref{erwgwegergewrwferwerc} we see that  it suffices to show that \begin{equation}\label{qwefqwefwefewfqwf}
K^{G}_{N(A)}(E_{\mathbf{Fin}}G^{\mathrm{CW}})\simeq 0\ .
\end{equation}
  Since $N(A)$ belongs to $\mathcal{CC}$  we have $\mathrm{Res}^{G}_{H,s}(N(A))\simeq 0$   for all $H$ in  $\mathbf{Fin}$. As a consequence of
 \eqref{fvdsfvsavsdcasdcasdc}   we have
 $K^{G}_{N(A)}(G/H)\simeq 0$ for all $H$ in $\mathbf{Fin}$.
 On the other hand, by the characterization of the homotopy type of $E_{\mathbf{Fin}}G^{\mathrm{CW}}$ we have    $Y^{G}(E_{\mathbf{Fin}}G^{\mathrm{CW}})(G/H)\simeq 0$  (see \eqref{wefqwefweqdxasdc} for $Y^{G}$) provided $H\not\in \mathbf{Fin}$.
 As an immediate consequence of   the formula  \eqref{qwefpojfopwefqwefqwefqew} for the evaluation of a homology theory on a $G$-topological space we get the desired equivalence \eqref{qwefqwefwefewfqwf}.
\end{proof}

Let $A$ be in $\mathrm{KK}_{\sepa}^{G}$.
\begin{lem}\label{egiowgwgeregegwref}
If $A$ is in $\mathcal{CI}$, then the Davis-L\"uck assembly map $\mu^{DL}_{A}$
is an equivalence.
\end{lem}
\begin{proof} 
Since $ K^{G}$ preserves countable colimits and $\mathcal{CI}$ is generated by $\Ind_{H,s}^{G}(B)$ for all $B$ in $\mathrm{KK}_{\sepa}^{G}$ and $H$ a finite subgroup of $G$ it suffices to show that 
 $\mu^{DL}_{\Ind_{H,s}^{G}(B)}$ is an equivalence for such data. 
 By \eqref{cdscasdc} we have an equivalence
 $K^{G}_{\Ind_{H,s}^{G}(B)}\simeq i^{G}_{H,!} K^{G}_{B}$.
 It is now a general fact (see e.g. \cite[Lemma 17.31]{cank} for an argument) that for  a functor $E:H\mathbf{Orb}\to \mathbf{M}$ with cocomplete stable target $\mathbf{M}$  we have a natural equivalence of functors
 $$i_{H,!}^{G}E\simeq E\circ \mathrm{Res}^{G}_{H}:G\Top\to \mathbf{M}\ ,$$
  where $i_{H,!}^{G}E$ denotes the extension to $G\Top$ of the functor 
  $i_{H,!}^{G}E$ on $G\mathbf{Orb}$.
 We therefore get the commutative square
  $$\xymatrix{K^{G}_{\Ind_{H,s}^{G}(B)}(E_{\mathbf{Fin}}G^{\mathrm{CW}}) \ar[rr]^-{\mu^{DL}_{\Ind_{H,s}^{G}(B)}}\ar[d]^{\simeq}&& K^{G}_{\Ind_{H,s}^{G}(B)}(*) \ar[d]^{\simeq}\\  
  K^{H}_{ B}(\mathrm{Res}^{G}_{H}(E_{\mathbf{Fin}}G^{\mathrm{CW}}))
  \ar[rr]^-{!}&&K^{H}_{ B}(\mathrm{Res}^{G}_{H}(*))}$$
  Since  
 $\mathrm{Res}^{G}_{H}(E_{\mathbf{Fin}}G^{\mathrm{CW}})\to \mathrm{Res}^{G}_{H}(*)$ is a homotopy equivalence in $H\Top$ we conclude that the  map marked by $!$ is an equivalence.
 This implies that $\mu^{DL}_{\Ind_{H,s}^{G}(B)}$ is an equivalence. 
   \end{proof}
   This finishes the proof of Theorem \ref{qerighojwergerfewferf}.
 \end{proof}
     

 
 We now discuss the construction of the functor $K^{G}$. {It is based on the ideas of Kranz \cite{kranz}, but we  reformulate the construction such that it only uses the formal aspects of the calculus of equivariant $\mathrm{KK}$-theory as developed in \cite{KKG}.
 We give full details since we use them   crucially  in the argument for  Proposition \ref{weroigjoewrgergergfewrf}, which in turn is used in Proposition \ref{eqriogergreqfqwfefq}.
 }
 

We start with the adjunction
\begin{equation}\label{ewrfewrfwefefewfefwefe}
\C[-]:G\Set\leftrightarrows \Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}}):\mathrm{Ob}
\end{equation} 
whose left adjoint sends a $G$-set  $S$ to the $G$-category $\C[S]$ with the $G$-set $S$ of objects   and morphisms generated by the identities \cite[Lem. 3.8]{crosscat}.  
By \cite[Lem. 3.7]{crosscat} the inclusion 
$ \Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})\to  \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ is again a left-adjoint. 
 By post-composition   with this inclusion we therefore get a left-adjoint functor $$\C[-]:G\Set\to \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$$ which we denote by the same    same symbol for simplicity. 

We now consider the functor
\begin{eqnarray}
\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\times G\Set& \stackrel{\id_{ \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})}\times \C[-]}{\to}& \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\times  \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}}) \nonumber\\&\stackrel{-\otimes_{\max}-}{\to}&  \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}}) \nonumber\\&\stackrel{\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}}{\to}&\mathrm{KK}^{G}\, ,\label{trhoiwrhrhehrthh} 
\end{eqnarray}
or  equivalently, using the exponential law, the functor
$$R^{G}:\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\to \Fun(G\Set,\mathrm{KK}^{G})\, , \quad \bC\mapsto R^{G}_{\bC}\, .$$
Let $i_{\omega}:G\Set_{\omega}\to G\Set$ denote the inclusion of the full subcategory of countable $G$-sets, and let $\Fun^{\coprod_{\omega}}$ the denote the full subcategory of a functor category of countable coproduct preserving functors. Let $y^{G}:\mathrm{KK}_{\sepa}^{G}\to \mathrm{KK}^{G}$ be as in \cite[Def. 1.8]{KKG}.
\begin{lem}\label{wegoiwegergrweg}
\mbox{}
\begin{enumerate}
\item \label{wethwthgegergr}$R^{G}$ is $s$-finitary.
\item \label{wethwthgegergr1}The restriction of $R^{G}$ to $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})$ sends unitary equivalences to equivalences.
\item \label{wethwthgegergr2}The functor $R^{G}$  sends weak Morita equivalences to equivalences.
\item \label{wethwthgegergr3}We have a canonical factorization
$$\xymatrix{\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})\ar[r]^-{\mathrm{kk}^{G}}\ar[d]^{\mathrm{incl}}&\mathrm{KK}^{G}\ar@{..>}[d]^{F^{G}}\\\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\ar[r]^-{R^{G}}& \Fun(G\Set,\mathrm{KK}^{G})}$$
\item  \label{wethwthgegergr4}The functor $F^{G}$  is exact and preserves    colimits.
\item \label{wethwthgegergr5} We have a factorization \begin{equation}\label{afdvoihvioasvsdsdvsavasdvds}
\xymatrix{\mathrm{KK}_{\sepa}^{G}\ar[r]^{y^{G}}\ar@{..>}[dd]^{F^{G}_{s}}&\mathrm{KK}^{G}\ar[d]^{F^{G}}\\&\Fun(G\Set,\mathrm{KK}^{G})\ar[d]^{i_{\omega}^{*}}\\ \Fun^{\coprod_{\omega}}(G\Set_{\omega},\mathrm{KK}_{\sepa}^{G}) \ar[r]^{y^{G}}&\Fun(G\Set_{\omega},\mathrm{KK}^{G}) }
\end{equation}such that $F_{s}$ is exact and preserves countable colimits.
\end{enumerate}
\end{lem}
\begin{proof}
Using the fact that $\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ is symmetric monoidal  \cite[Thm. 1.31]{KKG} we can rewrite the  functor  in  \eqref{trhoiwrhrhehrthh} as 
 \begin{equation}\label{fqwefqefdqewd}
 \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})\times G\Set   \stackrel{ \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}\times \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\C[-])}{\to} \mathrm{KK}^{G}\times \mathrm{KK}^{G} \stackrel{-\otimes_{\max}-}{\to}   \mathrm{KK}^{G}\ .
\end{equation}
 
The Assertions  \ref{wethwthgegergr}, \ref{wethwthgegergr1} and \ref{wethwthgegergr2} now follow from the corresponding properties of the functor
$\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}$ stated in \cite[Thm. 1.29]{KKG}, where  for  \ref{wethwthgegergr} we also use that
 the tensor structure on $\mathrm{KK}^{G}$ preserves colimits in each variable. 
 In order to show Assertion \ref{wethwthgegergr3} we again use the formula \eqref{fqwefqefdqewd}. It is then clear that we must define $F^{G}$ by the composition
 \begin{align}\label{wergfrfwefweerf}\\
 \mathclap{
F^{G}:\mathrm{KK}^{G}\stackrel{\id_{\mathrm{KK}^{G}}\times \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\C[-])}{\to} \mathrm{KK}^{G}\times \Fun(G\Set,\mathrm{KK}^{G})\stackrel{-\otimes_{\max}-}{\to} \Fun(G\Set,\mathrm{KK}^{G})\ , \quad A\mapsto F^{G}_{A}
}\notag
\end{align} 
 Since $-\otimes_{\max}-$ is bi-exact and preserves colimits in each argument  we conclude Assertion \ref{wethwthgegergr4}.
 
 
 
 We finally show Assertion \ref{wethwthgegergr5}.   We define
 $F^{G}_{s}$ by the formula \begin{equation}\label{ewfqwefwdqwedwqwed}
F^{G}_{s,(-)}(-)\simeq (-)\otimes_{\max} \mathrm{kk}_{\sepa}^{G}(A^{f}(\C[i_{\omega}(-)]))\ .
\end{equation}
   The following chain of equivalences yields the commutative square \eqref{afdvoihvioasvsdsdvsavasdvds}, where for the moment we ignore the superscript
   $\coprod_{\omega}$ at the lower left corner
  \begin{align*}
  \mathclap{
  y^{G}\circ F^{G}_{(-),s}( -)\stackrel{def}{\simeq} 
   y^{G}( (-)\otimes_{\max} \mathrm{kk}_{\sepa}^{G}(A^{f}(\C[-])))\stackrel{!}{\simeq} 
   y^{G}(-)\otimes_{\max}  \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G} (\C[i_{\omega}(-)])))\stackrel{def}{\simeq}  F^{G}_{y^{G}(-)}(i_{\omega}(-))\,.
   }
   \end{align*}
   For  the marked equivalence we use that $y^{G}$ is symmetric monoidal and the obvious equivalence $y^{G}( \mathrm{kk}_{\sepa}^{G}(A^{f}(\C[-])))\simeq \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\C[i_{\omega}(-)])$
   of functors from $G\Set_{\omega}$ to $\mathrm{KK}^{G}$.
 
   It remains to show that for any $A$ in $\mathrm{KK}_{\sepa}^{G}$ the functor
   $F_{s,A}^{G}$ preserves countable coproducts.
By definition,  we have  an equivalence
   $$F^{G}_{s,A}(-)\stackrel{def}{\simeq} 
 A\otimes_{\max} \mathrm{kk}_{\sepa}^{G}(A^{f}(\C[i_{\omega}( -)])) $$
      of functors from $G\Set_{\omega}$ to $\mathrm{KK}_{\sepa}^{G}$.

The functor $i_{\omega}$ preserves countable coproducts, and 
   since $\C[-]$ is a left-adjoint   
 it preserves all coproducts. The functor 
 $  \mathrm{kk}_{\sepa}^{G}\circ A^{f}$  sends the relevant countable coproducts to sums by  
 \cite[Lem. 6.6.]{KKG}.  Finally, by \cite[Prop. 1.7]{KKG}
 the tensor product $-\otimes_{\max}-$ on $\mathrm{KK}_{\sepa}^{G}$ preserves countable sums in each argument. This finishes the construction of the factorization $F^{G}_{s}$ asserted in \ref{wethwthgegergr5}.
 
 
 

 
 It immediately follows from the definition in \eqref{ewfqwefwdqwedwqwed} that 
 the functor $F_{s}^{G}$ is exact and preserves countable colimits.
 Here we use again that $-\otimes_{\max}-$  on $\mathrm{KK}_{\sepa}^{G}$ is bi-exact and preserves countable colimits in each argument\cite[Prop. 1.7]{KKG}.
This finishes the verification of Assertion  \ref{wethwthgegergr5}.  
  \end{proof}


  
 
 
  Let $H$ be a subgroup of $G$ and consider the object $G/H$ in $G\Set$. We let
$r^{G}_{H}:G\Set\to H\Set$ denote  the functor which restricts the $G$-action on a set to a $H$-action.
We consider  the object $G/H$ in $G\Set$.
\begin{lem}\label{ewgoweprgergwergwerg}
\mbox{}
\begin{enumerate}
\item \label{ergwioeogewgergergewrg}We have a commutative square  \begin{equation}\label{qwfeqweqfqwdqdqwed}
\xymatrix{\mathrm{KK}^{G}\ar[d]^{\mathrm{Res}^{G}_{H}}\ar[r]^-{F^{G}}&\Fun(G\Set,\mathrm{KK}^{G}) \ar[d]^{\ev_{G/H}}\\ \mathrm{KK}^{H}\ar[r]^{\Ind_{H}^{G}}&\mathrm{KK}^{G}}
\end{equation}
\item \label{wetgioweggwregwreg}We have a commutative square
 \begin{equation}\label{fewfqewfwfwefqewfqewf}
\xymatrix{\mathrm{KK}^{H}\ar[r]^-{F^{H}}\ar[dd]^{\Ind^{G}_{H}}&\Fun(H\Set,\mathrm{KK}^{H})\ar[d]^{r^{G,*}_{H}}\\ &\Fun(G\Set,\mathrm{KK}^{H})\ar[d]^{\Ind_{H}^{G}}\\\mathrm{KK}^{G}\ar[r]^-{F^{G}}&\Fun(G\Set,\mathrm{KK}^{G})}
\end{equation}
 \end{enumerate}
 \end{lem}
\begin{proof}
We use the functor $A:C^{*}\mathbf{Cat}^{\mathrm{nu}}\to C^{*}\mathbf{Alg}^{\mathrm{nu}}$ (see e.g. \cite[Def. 6.5]{crosscat}) and note that  for $S$ in $G\Set$ we get $A(\C[S])\cong C_{0}(S)$ in $\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$.  Applying this to $G/H$ in place of $S$ und using the definition of the induction functor
$\Ind_{H}^{G}$ from $\Fun(BH,C^{*}\mathbf{Alg}^{\mathrm{nu}})$ to $\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}})$
applied to $\C$ with the trivial $H$-action      we obtain the isomorphisms $$A(\C[G/H])\cong C_{0}(G/H)\cong \Ind_{H}^{G}(\C)\ .$$
By  \cite[Prop. 6.4]{KKG} for every $\bC$ in $ \Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$ have an equivalence  
$$\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\bC)\stackrel{def}{=}\mathrm{kk}^{G} ( A^{f}(\bC)) \stackrel{\alpha_{\bC}, \simeq}{\to}  \mathrm{kk}^{G}(A(\bC))\ .$$
Hence 
$$\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\C[G/H]) \simeq \mathrm{kk}^{G}(A(\C[G/H]))\simeq
\mathrm{kk}^{G}(\Ind_{H}^{G}(\C))\simeq \Ind_{H}^{G}(\mathrm{kk}^{G}(\C))\ $$
where the symbol $\Ind_{H}^{G}$ on the right-hand side is the induction functor from $\mathrm{KK}^{H}$ to $\mathrm{KK}^{G}$
\cite[Thm. 1.21]{KKG}.
Using \eqref{wergfrfwefweerf}, the following  projection formula  \cite[Cor. 4.10]{KKG} 
\begin{equation}\label{fwerfwfwfwefqwef}
\Ind_{H}^{G}(-)\otimes_{\max} (-)\simeq \Ind((-)\otimes_{\max} \mathrm{Res}^{G}_{H}(-))
\end{equation} 
for functors $\mathrm{KK}^{H}\times \mathrm{KK}^{G}\to \mathrm{KK}^{G}$ and that 
$\mathrm{kk}^{G}(\C)$ is the tensor unit of $\mathrm{KK}^{G}$ we get the following chain of equivalences    of endofunctors 
 \begin{eqnarray*}
\ev_{G/H}\circ F^{G}_{(-)}&\stackrel{}{\simeq} &( -) \otimes_{\max}  \Ind_{H}^{G}(\mathrm{kk}^{G}(\C))\\&\stackrel{}{\simeq}&
\Ind_{H}^{G}(\mathrm{Res}^{G}_{H} (-)\otimes_{\max}   \mathrm{kk}^{G}(\C))\\&\simeq&
\Ind_{H}^{G}(\mathrm{Res}^{G}_{H} (-))
\end{eqnarray*}
of $\mathrm{KK}^{G}$
which provides the filler of the square in \eqref{qwfeqweqfqwdqdqwed}.

 



 
In order to construct the filler of the pentagon in \eqref{fewfqewfwfwefqewfqewf}
we note that we have obvious equivalences
\begin{align}\label{qewfqwfewfwfqwdewdqwed}\\
\mathclap{
r^{G,*}_{H}(\mathrm{kk}_{C^{*}\mathbf{Cat}}^{H}(\C[-]))\simeq \mathrm{kk}_{C^{*}\mathbf{Cat}}^{H}(r^{G,*}_{H}(\C[-]))\simeq 
 \mathrm{kk}_{C^{*}\mathbf{Cat}}^{H}(\mathrm{Res}^{G}_{H}(\C[-]))\simeq \mathrm{Res}^{G}_{H}(\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\C[-]))\,.
 }\notag
\end{align} 
The chain of equivalences
 \begin{eqnarray*}
 \Ind_{H}^{G}\circ r^{G,*}_{H}\circ F^{H}_{(-)}&\stackrel{\text{def}}{\simeq}&
 \Ind_{H}^{G}((-)\otimes_{\max}  r^{G,*}_{H}(\mathrm{kk}_{C^{*}\mathbf{Cat}}^{H}(\C[-])))\\&\stackrel{\eqref{qewfqwfewfwfqwdewdqwed}}{\simeq}&
  \Ind_{H}^{G}(-\otimes_{\max}  \mathrm{Res}^{G}_{H} (\mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\C[-])))\\
  &\stackrel{\eqref{fwerfwfwfwefqwef}}{\simeq}&
  \Ind_{H}^{G}(-)\otimes_{\max}  \mathrm{kk}_{C^{*}\mathbf{Cat}}^{G}(\C[-])\\&\stackrel{\text{def}}{\simeq}&F^{G}_{(-)}\circ \Ind_{H}^{G}
\end{eqnarray*}
 provides the filler of   the pentagon.
\end{proof}



We now define the functor 
$$L^{G}:\mathrm{KK}^{G}\stackrel{F^{G}}{\to} \Fun(G\Set,\mathrm{KK}^{G})\stackrel{-\rtimes_{r}G}{\to}
\Fun(G\Set,\mathrm{KK})\ .$$
By \cite[Lem. 4.12 ]{KKG} {the} restriction of $-\rtimes_{r}G$ to the subcategories of compact objects is a countable  sum preserving functor functor 
$$(-\rtimes_{r}G)_{s}:\mathrm{KK}_{\sepa}^{G}\to \mathrm{KK}_{\sepa}\ .$$
We can therefore also define
$$L^{G}_{s}:\mathrm{KK}_{\sepa}^{G}\stackrel{F_{s}^{G}}{\to} \Fun^{\coprod_{\omega}}(G\Set_{\omega},\mathrm{KK}_{\sepa}^{G})\stackrel{-\rtimes_{r}G}{\to}
\Fun^{\coprod_{\omega}}(G\Set_{\omega},\mathrm{KK})\ .$$



\begin{lem}\label{qerigowegwergrewf}\mbox{}
\begin{enumerate}
\item \label{ewrgoiwjogegergwegr}$L^{G}$ is exact and preserves colimits.
\item  \label{ewrgoiwjogegergwegr1} For every subgroup $H$ of $G$ we have a commutative  square  \begin{equation}\label{ergwerrewgergregwerg}
\xymatrix{\mathrm{KK}^{H}\ar[r]^-{L^{H}}\ar[d]^{\Ind_{H}^{G}}& \Fun(H\Set,\mathrm{KK})\ar[d]^{r^{G,*}_{H}}\\\mathrm{KK}^{G}\ar[r]^-{L^{G}}& \Fun(G\Set,\mathrm{KK})}
\end{equation}
\item \label{rthkoperhrtehgrteg}For every subgroup $H$ of $G$ we have a commutative square
\begin{equation}\label{foijeovfvjkourf98adsfik}
\xymatrix{\mathrm{KK}^{G}\ar[d]^{\mathrm{Res}^{G}_{H}}\ar[r]^-{L^{G}}&\Fun(G\Set,\mathrm{KK})\ar[d]^{\ev_{G/H}}\\\mathrm{KK}^{H}\ar[r]^{-\rtimes_{r}H}&\mathrm{KK} }
\end{equation}
  \item \label{ewrgoiwjogegergwegr2} We have a commutative diagram
 \begin{equation}\label{afdvoihvioasvsdsdvsavasdvd2e2e2e2es}
\xymatrix{\mathrm{KK}_{\sepa}^{G}\ar[r]^{y^{G}}\ar[dd]^{L^{G}_{s}}&\mathrm{KK}^{G}\ar[d]^{L^{G}}\\&\Fun(G\Set,\mathrm{KK})\ar[d]^{i_{\omega}^{*}}\\ \Fun^{\coprod_{\omega}}(G\Set_{\omega},\mathrm{KK}_{\sepa}) \ar[r]^{y}&\Fun(G\Set_{\omega},\mathrm{KK}) }
\end{equation}
\item \label{ewrgoiwjogegergwegr3} For every subgroup $H$ of $G$ we have a commutative  square  \begin{equation}\label{ergwerrewgergregwdsfasdfsferg}
\xymatrix{\mathrm{KK}_{\mathrm{sep}}^{H}\ar[r]^-{L_{s}^{H}}\ar[d]^{\Ind_{H,s}^{G}}& \Fun^{\coprod_{\omega}}(H\Set_{\omega},\mathrm{KK}_{\sepa})\ar[d]^{r^{G,*}_{H}}\\\mathrm{KK}_{\sepa}^{G}\ar[r]^-{L_{s}^{G}}& \Fun^{\coprod_{\omega}}(G\Set_{\omega},\mathrm{KK}_{\sepa})}
\end{equation}
\item  \label{tgokwpegwergergeg}The functor $L_{s}$ is exact and preserves countable colimits.

\end{enumerate}

\end{lem}
\begin{proof}
Assertion \ref{ewrgoiwjogegergwegr} follows from   \ref{wegoiwegergrweg}.\ref{wethwthgegergr4} and the fact that $-\rtimes_{r}G:\mathrm{KK}^{G}\to \mathrm{KK}$ is exact and preserves colimits \cite[Thm.\ 1.21]{KKG}.
 
 For Assertion \ref{ewrgoiwjogegergwegr1} we expand the square in \eqref{ergwerrewgergregwerg} as follows:
  \begin{equation}\label{ergwerrewgergregwerg1}
\xymatrix{\mathrm{KK}^{H}\ar[r]^-{F^{H}}\ar[dd]^{\Ind_{H}^{G}}\ar[r]& \Fun(H\Set,\mathrm{KK}^{H})\ar[d]^{r^{G,*}_{H}}\ar[r]^{\rtimes_{r}H}& \Fun(H\Set,\mathrm{KK})\ar[dd]^{r^{G,*}_{H}}\\& \Fun(G\Set,\mathrm{KK}^{H})\ar[d]^{\Ind_{H}^{G}}\ar[dr]^{-\rtimes_{r}H}&\\\mathrm{KK}^{G}\ar[r]^-{F^{G}}& \Fun(G\Set,\mathrm{KK}^{G})\ar[r]^{-\rtimes_{r}G}& \Fun(G\Set,\mathrm{KK})}
\end{equation}
The left pentagon is precisely \eqref{fewfqewfwfwefqewfqewf} and commutes by   \ref{qerigowegwergrewf}.\ref{wetgioweggwregwreg}. The upper right square in \eqref{ergwerrewgergregwerg1} commutes by the associativity of composition of functors.
Finally, the lower triangle commutes by the equivalence 
\begin{equation}\label{erferfwerfwerfwerfe}
(-)\rtimes_{r}H\simeq \Ind_{H}^{G}(-)\rtimes_{r}G 
\end{equation}  of functors
from $\mathrm{KK}^{H}$ to $\mathrm{KK}$ \cite[Prop. 4.17]{KKG}.

In order to show Assertion \ref{rthkoperhrtehgrteg} expand the square \eqref{foijeovfvjkourf98adsfik} as follows:
\begin{equation}\label{foijeovfvjkourf98adsfike}
\xymatrix{\mathrm{KK}^{G}\ar[d]^{\mathrm{Res}^{G}_{H}}\ar@/^1cm/[rr]^{L^{G}}\ar[r]^-{F^{G}}&\Fun(G\Set,\mathrm{KK}^{G})\ar[d]^{\ev_{G/H}} \ar[r]^{-\rtimes_{r}G}&\Fun(G\Set,\mathrm{KK})\ar[d]^{\ev_{G/H}}\\\mathrm{KK}^{H}\ar@/^-1cm/[rr]_{-\rtimes_{r}K}\ar[r]^{\Ind_{H}^{G}}&\mathrm{KK}^{G}\ar[r]^{-\rtimes_{r}G}&\mathrm{KK} }
\end{equation}
The right square commutes obviously, and the commutativity of the left square is considered in \ref{ewgoweprgergwergwerg}.\ref{ergwioeogewgergergewrg}.
The upper triangle reflects the definition of $L^{G}$, and the lower triangle commutes 
by \eqref{erferfwerfwerfwerfe}.

Assertion \ref{ewrgoiwjogegergwegr2} follows from \ref{wegoiwegergrweg}.\ref{wethwthgegergr5} by composing with $-\rtimes_{r}G$ and the equivalence
$$(-\rtimes_{r}G)\circ y^{G}\simeq y\circ (-\rtimes_{r}G)_{s}$$ \cite[Lem.\ 4.12]{KKG}.
\end{proof}

The following discussion prepares the verification of Assertion  \ref{ewrgoiwjogegergwegr3}.
We have an adjunction
$$i^{G}_{H}:H\Set\leftrightarrows G\Set: r^{G}_{H}\ ,$$
where $i^{G}_{H}$ sends the $H$-set $S$ to the $G$-set $G\times_{H}S$.
Consequently, we have an equivalence 
 $$r^{G,*}_{H}\simeq i^{G}_{H,!}:\Fun(H\Set,\cC)\to \Fun(G\Set,\cC)$$ for any target category $\cC$, where $ i^{G}_{H,!}$ is the left Kan-extension functor.
The functor $i^{G}_{H}$ restricts to a functor
$i^{G}_{H}:H\mathbf{Orb}\to G\mathbf{Orb}$.
We note that the slice categories 
$H\mathbf{Orb}_{/S}$ for any $S$ in $G\mathbf{Orb}$ are   countable  discrete.  
Therefore the left Kan extension functor 
$$i^{G}_{H,!}:\Fun(H\mathbf{Orb},\cC)\to \Fun(G\mathbf{Orb},\cC)$$ exists provided $\cC$ admits countable  coproducts.
We let $i^{G}:G\mathbf{Orb}\to G\Set$ denote the inclusion.
From now on we assume that $\cC$ admits  countable coproducts.
We consider the square 
$$\xymatrix{\Fun(H\Set_{\omega},\cC)\ar[r]^{i^{H,*}}\ar[d]^{r^{G,*}_{H}}&\Fun(H\mathbf{Orb},\cC)\ar[d]^{i^{G}_{H,!}}\\ \Fun(G\Set_{\omega},\cC)\ar[r]^{i^{G,*}}&\Fun(G\mathbf{Orb},\cC)}$$
In general we do not expect that the square commutes.  
 \begin{lem}\label{wetigowegerferfwerf}
The restriction of the square to countable coproduct preserving functors
is a commutative square 
\begin{equation}\label{wegwergfergerwgrgf}
\xymatrix{\Fun^{\coprod_{\omega}}(H\Set_{\omega},\cC)\ar[r]_{\simeq}^{i^{H,*}}\ar[d]^{r^{G,*}_{H}}&\Fun(H\mathbf{Orb},\cC)\ar[d]^{i^{G}_{H,!}}\\ \Fun^{\coprod_{\omega}}(G\Set_{\omega},\cC)\ar[r]_{\simeq}^{i^{G,*}}&\Fun (G\mathbf{Orb},\cC)}
\end{equation}
\end{lem}
\begin{proof}
The inverse of the horizontal arrows are the left Kan-extension functors 
along $i^{H}$ and $i^{G}$, respectively. Since we have a canonical isomorphism 
 Since $ i^{G}_{H}\circ i^{H}\cong  i^{G}\circ i^{G}_{H}$ of functors from $H\mathbf{Orb}$ to $G\Set$ the  square
 \begin{equation}\label{ergwergergerfwewrfefref}
\xymatrix{\Fun^{\coprod_{\omega}}(H\Set_{\omega},\cC) \ar[d]^{i^{G}_{H,!}}&\ar[l]^{\simeq}_{i^{H}_{!}}\Fun(H\mathbf{Orb},\cC)\ar[d]^{i^{G}_{H,!}}\\ \Fun^{\coprod_{\omega}}(G\Set_{\omega},\cC) &\ar[l]^{\simeq}_{i^{G}_{!}}\Fun (G\mathbf{Orb},\cC)}
\end{equation}
commutes. We obtain \eqref{wegwergfergerwgrgf} from \eqref{ergwergergerfwewrfefref} by inverting the horizontal arrows.


In order to show Assertion \ref{ewrgoiwjogegergwegr3} we precompose
the square in  \eqref{ergwerrewgergregwerg} with $y^{H}$ and $y^{G}$, respectively,  and restrict the results to countable sets. We use that $\Ind_{H}^{G}\circ y^{H}\simeq \Ind_{H,s}^{G}\circ y^{H}$.
This gives the outer square in 
 \begin{equation}
 \label{ergwerrewgergregwdsfascdcdcdfsferg}
\xymatrix{
\mathrm{KK}_{\mathrm{sep}}^{H} \ar@/^1cm/[rr]^{(L^{H}_{ y^{H}})_{|H\Set_{\omega}}}\ar[r]^-{L_{s}^{H}}\ar[d]^{\Ind_{H,s}^{G}}& \Fun^{\coprod_{\omega}}(H\Set_{\omega},\mathrm{KK}_{\sepa})\ar[d]^{r^{G,*}_{H}}\ar[r]^{y}&\Fun(H\Set_{\omega},\mathrm{KK})\ar[d]^{r^{G,*}_{H}}\\
\mathrm{KK}_{\sepa}^{G}\ar[r]^-{L_{s}^{G}}\ar@/_1cm/[rr]_{(L^{G}_{y^{G}})_{|G\Set_{\omega}}}& \Fun^{\coprod_{\omega}}(G\Set_{\omega},\mathrm{KK}_{\sepa})\ar[r]^{y}&\Fun(G\Set_{\omega},\mathrm{KK})}
\end{equation}

 We then use that
$r^{G}_{H}$ preserves countability and coproducts and therefore that  $r^{G,*}_{H}$  preserves countable  coproduct preserving functors. If we now employ the fact that    $y$ is fully faithful, then we get the filler of the left square.

Assertion \ref{tgokwpegwergergeg} follows from  \ref{qerigowegwergrewf}.\ref{wethwthgegergr5} and the fact that $(-\rtimes _{r}G)_{s}$ is exact und  preserves countable colimits \cite[Lemma 4.8]{KKG}.
 \end{proof}

Let $H$ be a subgroup of $G$.
Note that $\mathrm{KK}_{\sepa}$ admits countable colimits by \cite[Thm. 1.4]{KKG}.
\begin{prop}We have a commutative square \begin{equation}\label{wqoiwehfiwqeofwefwefweq}
\xymatrix{\mathrm{KK}_{\mathrm{sep}}^{H}\ar[r]^-{i^{H,*}L^{H}_{s}}\ar[d]^{\Ind_{H,s}^{G}}& \Fun(H\mathbf{Orb},\mathrm{KK}_{\sepa})\ar[d]^{i^{G}_{H,!}}\\\mathrm{KK}_{\sepa}^{G}\ar[r]^-{i^{G,*}L^{G}_{s}}&\Fun(G\mathbf{Orb},\mathrm{KK}_{\sepa})}
\end{equation}
 \end{prop}
\begin{proof}
We expand the square as 
$$\xymatrix{\mathrm{KK}_{\mathrm{sep}}^{H}\ar[r]^-{i^{H,*} }\ar[d]^{\Ind_{H,s}^{G}} \ar[d]&\Fun^{\coprod_{\omega}}(H\Set_{\omega},\mathrm{KK}_{\sepa})\ar[d]^{r^{G,*}_{H}}\ar[r]^{L^{H}_{s}}& \Fun(H\mathbf{Orb},\mathrm{KK}_{\sepa})\ar[d]^{i^{G}_{H,!}}\\\mathrm{KK}^{G}\ar[r]^-{i^{G,*} }&\ar[r]^{L^{G}_{s}}\Fun^{\coprod_{\omega}}(G\Set_{\omega},\mathrm{KK}_{\sepa})&\Fun(G\mathbf{Orb},\mathrm{KK}_{\sepa})}$$
The left square commutes by  \ref{qerigowegwergrewf}.\ref{ewrgoiwjogegergwegr3}.
The right square commutes by Lemma \ref{wetigowegerferfwerf}.
\end{proof}



\begin{ddd}\label{egiueheiugheiugwegerggwfewr}
We define the functors
$$\hat K^{G}:\mathrm{KK}^{G}\stackrel{i^{G,*}L^{G}}{\to} \Fun(G\mathbf{Orb},\mathrm{KK}^{G})\stackrel{\mathrm{KK}^{G}(\C,-)}{\to} \Fun(G\mathbf{Orb},\Sp)$$
and 
$$K^{G}:=\hat K^{G}\circ y^{G}:{\mathrm{KK}_{\sepa}^{G}}\to  \Fun(G\mathbf{Orb},\Sp)\ .$$
\end{ddd}
Using the equivalence
$\mathrm{KK}^{G}(\C,y^{G}(-))\simeq \mathrm{KK}_{\sepa}^{G}(\C,-)$ of functors from $\mathrm{KK}_{\sepa}$ to $\Sp$ we get the formula
\begin{equation}\label{qwefweqdqwedqw}
K^{G}\simeq \mathrm{KK}_{\sepa}(\C,i^{G,*}L^{G}_{s}(-))\ .
\end{equation}
\begin{theorem}
The functor $K^{G}$ satisfies the Assumption \ref{tgioewrgergwerg}.
\end{theorem}
\begin{proof}
The functor $K^{G}$ is exact since $i^{G,*}L^{G}$ is exact by  \ref{qerigowegwergrewf}.\ref{ewrgoiwjogegergwegr}
 and the fact that $\mathrm{KK}^{G}(\C,-)$ and $y^{G}$ are  exact.
 
In order to show that the functor $K^{G}$ preserves countable colimits we use \eqref{qwefweqdqwedqw}, that  $L_{s}^{G}$
preserves countable colimits by \ref{qerigowegwergrewf}.\ref{tgokwpegwergergeg},
and that  $ \mathrm{KK}_{\sepa}^{G}(\C,-)$ preserves  countable colimits.
Indeed, $\mathrm{KK}_{\sepa}^{G}(\C,-)$ is exact by definition. In order to see that it preserves countable sums, we use   the identification $ \mathrm{KK}_{\sepa}^{G}(\C,\mathrm{kk}_{\mathrm{sep}}(-))\simeq K^{C^{*}\mathrm{Alg}}(-)$ of functors from $C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}\to \Sp$, the fact that countable sums in $\mathrm{KK}_{\sepa}$ are presented by countable sums in $C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}}$, and that 
$K^{C^{*}\mathrm{Alg}}$ sends countable sums to coproducts. 

For $A$ in $\mathrm{KK}_{\sepa}^{G}$ we have a natural equivalence
\begin{eqnarray*}
K^{G}_{A}(G/H)&\simeq&\mathrm{KK}(\C, L^{G}_{y^{G}(A)}(G/H))\\
&\stackrel{\ref{qerigowegwergrewf}.\ref{rthkoperhrtehgrteg}}{\simeq}&
\mathrm{KK}(\C,  \mathrm{Res}^{G}_{H}(A)\rtimes_{r}H)\ .
\end{eqnarray*}
Finally the commutative of the square in \eqref{cdscasdc}  is obtained by applying $\mathrm{KK}_{\sepa}(\C,-)$ to the square in \eqref{wqoiwehfiwqeofwefwefweq} and  
using that $\mathrm{KK}_{\sepa}(\C,-):\mathrm{KK}_{\sepa}\to \Sp$ preserves countable colimits in order to commute
$i^{G}_{H,!}$ with this functor.
 \end{proof}

 


We consider $A$ in $\Fun(BG,C^{*}\mathbf{Alg}^{\mathrm{nu}}_{\mathrm{sep}})$ and  form the pair  $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}_{c}^{G}(A))$ of a unital  {$G$}-$C^{*}$-category with an invariant ideal. Then we have the functor
$K\bC^{G}:G\mathbf{Orb}\to  \Sp$  introduced  in Definition 
\ref{qroeigjoqergerqfewewfqewfeqwf} and the functor $K^{G}_{A}:G\mathbf{Orb}\to \Sp$ given by Definition \ref{egiueheiugheiugwegerggwfewr}, where we simplify the notation by using $A$ instead of $\mathrm{kk}_{\sepa}^{G}(A)$  in order to indicate the dependence on $A$.


Note that the functor $K\bC^{G}$ is well-suited for the approach to the assembly map via Paschke duality
and coarse geometry. On the other hand, 
  $K^{G}_{A}$  is good if one is interested in the functorial  
   dependence  on $A$, a fact which is relevant for the approach to the assembly in \cite{kranz}.


\begin{prop}\label{weroigjoewrgergergfewrf}
If $A$ is unital, then there is a canonical equivalence
$K\bC^{G}\simeq K^{G}_{A}$.
\end{prop}
\begin{proof}
We start with recalling the explicit description of  the functor 
$$  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}((-)_{min,max}\otimes G_{can,min}):G\mathbf{Orb}\to {\mathbf{C}^{\ast}\mathbf{Cat}}$$ underlying the definition of $K\bC^{G}$ obtained in \cite[Prop. 8.2.4.]{coarsek}:
\begin{equation}\label{fewrfrefwerfweferfwefefref}
  \mathbf{\bar C}_{\mathrm{lf}}^{ G,\mathrm{ctr}}((-)_{min,max}\otimes G_{can,min})\cong {}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[-]\rtimes_{r}G\ .
\end{equation} 
{Here the functor $  {}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[-]:G\mathbf{Orb}\to \Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})$} is defined in \cite[Def. 17.9]{cank}, \cite[Def. 17.12]{cank}, and we use 
 \cite[Def. 17.12]{cank} in order  see that it is a well-defined {on morphisms}.
{Roughly, for $S$ in $G\mathbf{Orb}$  the objects of       ${}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[S]$         are objects of $\bC$ which are decomposed as orthogonal sums of objects from ${\mathbf{K}}^{u}$ indexed by  finite subsets of $S$. Morphisms are morphisms in $\bC$ which preserve this decomposition. The action of $G$ is induced from the actions on $\bC$
and the action on $S$ by shifting the index sets.
 The isomorphism \eqref{fewrfrefwerfweferfwefefref} induces}  an equivalence
\begin{equation}\label{wgrgrfwcercdsfvcsdcdcsd}
K\bC^{G}(-)\simeq  K^{C^{*}\mathrm{Cat}}({}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[-]\rtimes_{r}G)\ .
\end{equation} 
We now start from \eqref{trhoiwrhrhehrthh} in order to derive a similar formula for $K^{G}_{A}$.
We get \begin{equation}\label{fwefwefqfd}
K^{G}_{A}(-)\simeq  K^{C^{*}\mathrm{Cat}}( (A\otimes_{\max}\C[-])\rtimes_{r}G)\ .
\end{equation} We  define a natural transformation 
$$\phi:A\otimes_{\max}\C[-]\to {{}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[-]}$$
between functors from $G\mathbf{Orb}$ to $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Cat}})$.
The evaluation of $\phi$ at  $S$ in $G\mathbf{Orb}$ is inclusion  $\phi_{S}$ of a full subcategory which sends 
the object  $s$ in $S=\mathrm{Ob}(A\otimes_{\max}\C[S])$ to the object  $(A,p)$, where  the family of projections $p=(p_{s'})_{s'\in S}$ is given by
$$p_{s'}=\left\{\begin{array}{cc}1&s=s'\\0&s\not=s'\end{array}\right. $$
and we consider $A$ as an object of ${\mathbf{K}}^{u}$, which is possible by unitality of $A$.
{In other words, the object $\phi(s)$ is the object of ${}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[S]$
given by $A$ sitting on the single point $s$.}
 One checks that the functor is equivariant, and that the family
 $(\phi_{S})_{S\in G\mathbf{Orb}}$ is a natural transformation.
Note every object in ${ {}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[S]}$ is a summand of a finite sum of objects in the image of $\phi_{S}$. Indeed, if $(C,p)$ is supported on the finite subset $T$ of $S$, the  image of $p_{t}$ is in ${\mathbf{K}}^{u}$ and therefore a summand of $\bigoplus_{i=1}^{n_{t}}A$. Hence we find an isometric inclusion of
$(C,p)$ into $\bigoplus_{t\in T} \bigoplus_{i=1}^{n_{t}} \phi_{S}(t)$. 
We thus conclude that $\phi_{S}$ is a Morita equivalence for every $S$ on $G\mathbf{Orb}$.
 Since $-\rtimes_{r}G$ preserves Morita equivalences by \cite[Prop. 14.8]{cank}
 and $K^{C^{*}\mathrm{Cat}}$ sends Morita equivalences to equivalences
 we finally {conclude} that $\phi$ induces an equivalence  
\begin{equation}\label{fwefewfwdfwqdwedwedq}
 K^{C^{*}\mathrm{Cat}}( (A\otimes_{\max}\C[-])\rtimes_{r}G) \to  K^{C^{*}\mathrm{Cat}}({}_{{\mathbf{K}}}\bC_{\mathrm{lf}}[-]\rtimes_{r}G)\ .
\end{equation}  
 Combining the equivalences in \eqref{fwefwefqfd}, \eqref{wgrgrfwcercdsfvcsdcdcsd} and \eqref{fwefewfwdfwqdwedwedq} we get the desired equivalence.
\end{proof}




 
  
   {Let $A$ be in $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Alg}}_{\mathrm{sep}})$, {i.e.\ a unital separable $C^{*}$-algebra with a $G$-action.}}
\begin{prop}\label{eqriogergreqfqwfefq}
 If {$\mathrm{kk}_{\sepa}^{G}(A)$} is in $\mathcal{CI}$ and $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}^{G}_{c}(A))$, then the 
analytic assembly map $$\mathrm{Asmbl}^{\mathrm{an}}_{\mathbf{Fin}}\colon 
 RK^{G,\mathrm{An}}_{\bC}( E_{\mathbf{Fin}}G^{\mathrm{CW}})\to { \mathrm{KK}}(\C,{\mathbf{K}}^{{(G)}}_{\mathrm{std}}\rtimes_{r}G)$$ is an equivalence.
\end{prop}
\begin{proof}
It suffices to show that $\mathrm{Asmbl}^{\mathrm{an}}_{\cF}$ induces an isomorphism on the level of homotopy groups.
By Theorem \ref{wtoiguwegwergergregwe} we have an isomorphism
$\pi_{*+1}\mathrm{Asmbl}^{an}_{\mathbf{Fin}}\cong  \pi_{*}\mathrm{Asmbl}^{h}_{\mathbf{Fin}}$.
By Proposition \ref{weroigjoewrgergergfewrf} we have an  equivalence 
$\mathrm{Asmbl}^{h}_{\mathbf{Fin}}\simeq \mu^{DL}_{A}$. Finally, by 
 Lemma \ref{egiowgwgeregegwref} we know that 
  $\mu_{A}^{DL}$ is an equivalence. 
  The combination of these results yields the assertion.
  \end{proof}




Recall that we consider $A$ in $\Fun(BG,{\mathbf{C}^{\ast}\mathbf{Alg}}_{\mathrm{sep}})$ and $(\bC,{\mathbf{K}})=(\mathbf{Hilb}^{G}(A),\mathbf{Hilb}_{c}^{G}(A))$ a pair of a unital  $G$-$C^{*}$-category with an invariant ideal. 
The  following proposition  compares the assembly map $\mathrm{Asmbl}^{\mathrm{an}}_{\cF}$ associated with the pair $(\bC,{\mathbf{K}})$ with the   Kasparov assembly map $\mu^{Kasp}_{\cF ,A}$. Its second assertion is  the   statement   that
 the Kasparov assembly map is an  equivalence for compactly induced coefficients which has been used in the proof of Theorem \ref{wrigoghgegrtgeg} above. {This was shown more generally for locally compact groups $G$ in } \cite{MR1836047} . The proof of  the {following} proposition is logically independent of the results of  \cite{MR1836047}. In fact, one  can use  
  its second assertion in order to make the proof of  Theorem \ref{wrigoghgegrtgeg}
 independent of   \cite{MR1836047}. 
\begin{prop}\label{weiojgwoerferww}\mbox{}
\begin{enumerate}
\item If $\cF\subseteq \mathbf{Fin}$, then have a commutative square \begin{equation}\label{adfadfadfsafdsf}
\xymatrix{\Sigma RK^{G,\mathrm{an}}_{A}(E_{\cF}G^{\mathrm{CW}})\ar[r]^-{\Sigma \mu^{Kasp}_{\cF,A}}\ar@{-}[d]^{\simeq}&\Sigma  \mathrm{KK}(\C,A\rtimes_{r} G)\ar[d]\\ RK^{G,\mathrm{An}}_{\bC}( E_{\cF}G^{\mathrm{CW}}) \ar[r]_{\mathrm{Asmbl}^{\mathrm{an}}_{\cF}}&\mathrm{KK}(\C,{\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G)}
\end{equation}
 \item If $A$ is in $\mathcal{CI}$, then $$\mu^{Kasp}_{\mathbf{Fin},A}:RK^{G,\mathrm{an}}_{A}(E_{\mathbf{Fin}}G)\to   \mathrm{KK}( \C,A\rtimes_{r}G)$$ is an equivalence.
\end{enumerate}\end{prop}
\begin{proof}
We start with the construction of the square \eqref{adfadfadfsafdsf}.  
Its  left vertical morphism will be induced by a zig-zag and therefore does not have a preferred direction. We expand the square to the 
  following commutative diagram \begin{equation}\label{qewfqwefqwdqd}
\xymatrix{\Sigma RK^{G,\mathrm{an}}_{A}(E_{\cF}G^{\mathrm{CW}})\ar@/^1cm/[rr]^{\Sigma \mu^{Kasp}_{\cF,A}}\ar[r]^{\mu^{Kasp}_{\cF,A,\max}}\ar[d]^{\simeq} & \Sigma \mathrm{KK}(\C,A\rtimes G)\ar[r]\ar[d]^{\simeq}&\Sigma  \mathrm{KK}(\C,A\rtimes_{r} G)\ar[d]^{\simeq}\\\Sigma   RK^{G,\mathrm{an}}_{{\mathbf{K}}^{(G)}_{\mathrm{std}}}(E_{\cF}G^{\mathrm{CW}}) \ar[r]^{\Sigma \mu^{Kasp}_{\cF,{\mathbf{K}}^{(G)}_{\mathrm{std}},\max}}&\Sigma \mathrm{KK}(\C,{\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes  G) \ar[r]&\Sigma \mathrm{KK}(\C,{\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G) \ar@{=}[d] \\ RK^{G,\mathrm{an}}_{\mathbf{Q}^{(G)}_{\mathrm{std}}}(E_{\cF}G^{\mathrm{CW}}) \ar@{=}[d]\ar[r]^{\mu^{Kasp}_{\cF,\mathbf{Q}^{(G)}_{\mathrm{std}},\max}}\ar[u]^{\simeq}&\mathrm{KK}(\C,\mathbf{Q}^{(G)}_{\mathrm{std}}\rtimes G) \ar[r]\ar[u]^{\simeq}& \mathrm{KK}(\C,{\mathbf{K}}^{(G)}_{\mathrm{std}}\rtimes_{r} G) \\ RK^{G,\mathrm{An}}_{\bC}( E_{\cF}G^{\mathrm{CW}})\ar[rru]^{\simeq}_{\mathrm{Asmbl}^{\mathrm{an}}_{\cF}}&& }
\end{equation}
The upper vertical maps are induced by the canonical  inclusion $A\to {\mathbf{K}}^{(G)}_{\mathrm{std}}$ which exists since we assume that $A$ is unital. This inclusion is a weak Morita equivalence \cite[Ex. 16.13]{cank} and therefore induces equivalences as indicated, see the proof of Proposition \ref{wegojeogrregregwergwegre} for precise references.
 The upper right square
 is provided  by the    natural transformation $-\rtimes G\to -\rtimes_{r}G$.
The two lower left vertical arrows are induced by 
the  boundary map  of the fibre sequence 
 associated to the 
exact sequence  $0\to {\mathbf{K}}^{(G)}_{\mathrm{std}}\to  \bC^{(G)}_{\mathrm{std}}\to  \mathbf{Q}^{(G)}_{\mathrm{std}}\to 0$ in $\Fun(BG,C^{*}\mathbf{Cat}^{\mathrm{nu}})$, see the proof of Proposition \ref{wegojeogrregregwergwegre}. This connecting map is an equivalence
  since $\mathrm{kk}^{G}(\bC^{(G)}_{\mathrm{std}})\simeq 0$ since $\bC^{(G)}_{\mathrm{std}}$
is countably additive and therefore flasque.
The two left squares commute by the 
 naturality of the Kasparov assembly map with respect to the coefficients in $\mathrm{KK}^{G}$. 
 The upper triangle and the lower triangle reflect the Definitions \ref{wergoijowergerrrfrfrfrfeggwgw} and \ref{wergoijowergerreggwgw}  of
 $\mu_{\cF,A}^{Kasp}$ and $\mathrm{Asmbl}^{\mathrm{an}}_{\cF}$. 
 
 
 
 If   $\mathrm{kk}_{\sepa}^{G}(A)\in \mathcal{CI}$, and $\cF=\mathbf{Fin}$, then  by Proposition \ref{eqriogergreqfqwfefq}
the map 
 $\mathrm{Asmbl}^{\mathrm{an}}_{\mathbf{Fin}}$ is an equivalence. We conclude that 
  $\mu^{Kasp}_{\mathbf{Fin},A}$ is an equivalence, too.
\end{proof}


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