\section{Introduction}
Algebraic $kk$-theory was introduced in \cite{cortho}
 in order to show how methods from $K$-theory of operator
algebras can be applied in a completely algebraic setting. The definition of algebraic $kk$-theory was motivated by the works \cite{cuntz} and \cite{higson} on Kasparov's $KK$-theory introduced in \cite{kasp}. An equivariant version of the algebraic $kk$-theory was introduced in \cite{euge}. We can write a dictionary between Kasparov's $KK$-theory and algebraic $kk$-theory. Kasparov's $KK$-theory of separable $C^{*}$-algebras is the major tool in noncommutative topology. It is a common generalization both of topological $K$-homology and topological $K$-theory as an additive bivariant functor. Let $A$ and $B$ be separable $C^{*}$-algebras. The groups $KK_{*}(A,B)$ are defined such that
$$
KK_{*}(\mathbb{C},B)\cong K^{top}_{*}(B) \qquad\text{and} \qquad KK^{*}(A,\mathbb{C})\cong K_{hom}^{*}(A). 
$$
Here $K^{top}_{*}(B)$ denotes the topological $K$-theory groups of $B$ and $K_{hom}^{*}(A)$ the $K$-homology groups of $A$. 
The Kasparov groups $KK(A,B)=KK_{0}(A,B)$ for separable $C^{*}$-algebras $A$ and $B$ form the set of morphisms $A\rightarrow B$ in a category $KK$. The composition in $KK$ is given by the Kasparov product. The category $KK$ admits a triangulated category structure and there is a canonical functor $k:\mbox{$C^{*}$-$\operatorname{Alg}$}\rightarrow KK$ which is homotopy invariant, $C^{*}$-stable and split-exact. Moreover, $k$ is universal for these properties. Similarly 
algebraic $kk$-theory is a bivariant $K$-theory on ${\mathrm{Alg}_\ell}$, the category of algebras over a commutative ring $\ell$. For $\ell$-algebras $A$ and $B$ the groups $kk_{*}(A,B)$ are defined  such that
$$
kk_{*}(\ell, A) \cong \operatorname{KH}_{*}(A).
$$
Here $\operatorname{KH}_{*}(A)$ denotes Weibel's homotopy $K$-theory groups.
We can consider a triangulated category $kk$ whose objects are the $\ell$-algebras and whose morphisms are the elements of  $kk(A,B)=kk_{0}(A,B)$. There exists a canonical functor $j:{\mathrm{Alg}_\ell}\rightarrow kk$ such that $j$ is stable with respect to matrices, polynomial homotopy invariant and excisive. Moreover, $j$ is universal for these properties.    

Let $G$ be a (discrete) group. The equivariant Kasparov's $KK$-theory is a bivariant $K$-theory defined on $\mbox{$G$-$C^{*}$-$\operatorname{Alg}$}$, the category of separable $C^{*}$-algebras with an action by automorphisms of $G$. There exists a triangulated category $KK^{G}$ and a functor $j:\mbox{$G$-$C^{*}$-$\operatorname{Alg}$} \rightarrow KK^{G}$ that is universal for the properties: stable with respect to compact operators on $\ell^{2}(G\times \mathbb{N})$, continuous homotopy invariant and split exact. 
The equivariant algebraic $kk$-theory is a bivariant $K$-theory defined in ${G\mathrm{Alg}_\ell}$, the category of $\ell$-algebras with an action of $G$. There exists a functor $j:{G\mathrm{Alg}_\ell}\rightarrow kk^{G}$ universal for the properties: $G$-stable, polynomial homotopy invariant and excisive. In the  algebraic setting there are versions of the Green-Julg Theorem, the adjointness between induction and restriction functors and the Baaj-Skandalis duality. 
Let $K$ be a finite group whose order is invertible in $\ell$ and let $A\in {K\mathrm{Alg}_\ell}$. The Green-Julg Theorem gives us a natural isomorphism
\begin{equation}\label{gjmor}
\psi_{GJ}:kk^{K}_{*}(\ell, A) \xrightarrow{\cong} kk_{*}(\ell, A\rtimes K) =\operatorname{KH}_{*}(A\rtimes K).
\end{equation}
Let $H\leq G$. The adjunction between the induction and restriction functors gives us a natural isomorphism 
\begin{equation}\label{irmor}
\psi_{IR}:kk^{G}_{*}(\mathrm{Ind}^{G}_{H}(B),A)\xrightarrow{\cong} kk^{H}_{*}(B, {\mathrm{Res}}^{H}_{G}(A)), \qquad B\in {H\mathrm{Alg}_\ell}, A\in {G\mathrm{Alg}_\ell}.
\end{equation}
Let $H$ be a finite subgroup of $G$ with $\tfrac{1}{|H|} \in \ell$. Notice that $$\mathrm{Ind}^{G}_{H}(\ell)=\ell^{(G/H)}=\bigoplus_{G/H} \ell.$$ 
Combining the isomorphisms \eqref{gjmor} and \eqref{irmor} we obtain an isomorphism 
$$
\psi: kk^{G}_{*}(\ell^{(G/H)},A)\xrightarrow{\cong} \operatorname{KH}_{*}(A\rtimes H).
$$
Can we express $\operatorname{KH}_{*}(A\rtimes G)$ as a $kk^{G}$-group? If $G$ is finite with $\tfrac{1}{|G|}\in \ell$ the answer is positive; see \eqref{gjmor}. In this paper we go deep into this question. We consider the $\operatorname{KH}$-isomorphism conjecture introduced in \cite{balu} and we write the left hand side of this conjecture in terms of $kk^{G}$-groups. One motivation to do this is  to formulate in the algebraic context techniques used in some proofs of the Baum-Connes conjecture with coefficients.

The {\emph{Baum-Connes conjecture with coefficients}} is formulated in \cite{bacahi}. Let $G$ be a discrete group and let $A$ be a separable $G$-$C^{*}$-algebra. Let $E_{\mathcal{F}in}(G)$ denote a model for the classifying space of $G$ with respect to the family of finite subgroups.
The Baum-Connes property is true for the pair $(G,A)$ if the {\emph{assembly map}} $\mu_{G,A}$,
\begin{equation}\label{assmap}
\mu_{G,A}: KK_{*}^{G}(C_{0}(E_{\mathcal{F}in}(G)), A)=\operatornamewithlimits{colim}_{X}  KK_{*}^{G}(C_{0}(X), A)\rightarrow K^{top}_{*}( A\rtimes G),
\end{equation}
is an isomorphism; here $X$ runs over the directed set of $G$-compact subsets of $E_{\mathcal{F}in}(G)$.
For more details about the Baum-Connes conjecture see \cite{surbc}.

We are looking for an assembly map similar to \eqref{assmap} with $\operatorname{KH}$ instead of $K^{top}$ and algebraic $kk^{G}$-groups instead of Kasparov's $KK^{G}$-groups.

For discrete groups, the Baum-Connes conjecture can also be formulated using homotopy theory through the Davis-L\"uck approach \cite{dalu}.  
Write $\mathrm{Sp}$ for the category of (simplicial) spectra.
Let $\Or G$ denote the orbit category of $G$: its objects are $G/H$ with $H$ a subgroup of $G$ and its morphisms are $G$-equivariant maps  
$f: G/H\rightarrow G/K$.
Recall that an $\Or G$-spectrum $\bold{E}$ is a functor $\bold{E}:\Or G \rightarrow \mathrm{Sp}$.
In \cite{dalu}, an equivariant homology theory is constructed for each $\Or G$-spectrum $\bold{E}$. 
The homology theory $H^{G}(-;\bold{E})$ in the category of $G$-simplicial sets $\mathbb{S}^{G}$ is defined as follows:
\begin{equation}\label{forporciento}
H^{G}(-;\bold{E}): \mathbb{S}^{G} \rightarrow \mathrm{Sp}, \qquad H^{G}(X;\bold{E}) = \operatorname{map}_{G}(-,X)_{+}\otimes_{\Or G} \bold{E(-)}.
\end{equation}
Let $\mathcal{F}$ be a family of subgroups of $G$ and let $E_{\mathcal{F}}G$ be a model of the classifying space of $G$ with respect to $\mathcal{F}$, i.e. a $G$-simplicial set such that
$$
{(E_{\mathcal{F}}G)}^{H}= \left\{ \begin{array}{lll} \mbox{contractible} &\mbox{if}& \mbox{$H\in \mathcal{F}$,}\\
\emptyset & \mbox{if}& \mbox{$H \notin \mathcal{F}$.}\\
\end{array} \right.
$$
Let $p:E_{\mathcal{F}}(G)\rightarrow \star$ be the projection to the point.
The \emph{assembly map} associated to $(G,\mathcal{F},\bold{E})$ is the map 
\begin{equation}\label{asdl} H^{G}(p): H^{G}(E_{\mathcal{F}}(G);\bold{E})\rightarrow H^{G}(\star;\bold{E})\text{.} \end{equation}
We remark that in the original construction of \cite{dalu} topological spectra are used instead of simplicial spectra. In our work we use that $\mathrm{Sp}$ is a combinatorial category (see Lemma \ref{lem:sptCombi}) and that holds for simplicial spectra. In \cite{corel}*{Section 2} it is proved that $\operatorname{Top}^{G}$ is Quillen equivalent to $\mathbb{S}^{G}$ and that it is equivalent to  work with assembly maps in the topological or in the simplicial setting. 
If $\bold{E}$ is the topological spectrum $K^{top}_{A}$ and $\mathcal{F}=\mathcal{F}in$ then the assembly map \eqref{asdl} coincides with \eqref{assmap}, see \cite{kranz}. 

The $\operatorname{KH}$-isomorphism conjecture was introduced in \cite{balu}*{Section 7}; see \cite{bb}*{Section 15.3} for the status of this conjecture. In this paper we use a particular $\Or{G}$-spectrum $\bold{E}$ that we proceed to describe.
For a $G$-algebra $B$ and $G/H\in\Or{G}$, we define an algebra $\mathcal{R}(B\rtimes G/H)$ that has the properties of being natural in $G/H$ and $kk^G$-equivalent to $B\rtimes H$; see Sections \ref{sec:lincat} and \ref{sec:crospro}. Let $\mathbb{K}$ be the spectrum representing $kk$-theory defined by Garkusha \cite{garku}; see Section \ref{sec:sptkkg} for details.
The $\operatorname{KH}$-isomorphism conjecture states that for $\mathcal{F}=\mathcal{F}in$ and 
\begin{equation}\label{defe}
\bold{E}:\Or G \rightarrow \mathrm{Sp}\text{,} \quad \bold{E}(G/H)=\mathbb{K}(\ell, \mathcal{R}(B\rtimes G/H))\text{,}
\end{equation}
the assembly map  \eqref{asdl} is an isomorphism. Note that 
$$
\pi_{*}(\bold{E}(G/H))=\pi_{*}(\mathbb{K}(\ell, \mathcal{R}(B\rtimes G/H))= kk_{*}(\ell, \mathcal{R}(B\rtimes G/H))=\operatorname{KH}_{*}(B\rtimes H).
$$

Let $\mathbb{K}^G$ be the spectrum representing $kk^G$-theory defined in \cite{tesisema}; see Section \ref{sec:sptkkg} for details. The main theorem of this work is Theorem \ref{mainthm}:
\begin{thm}[cf. Theorem \ref{mainthm}]\label{thm:mainIntro}
Let $G$ be a group such that $\tfrac{1}{|H|}\in\ell$ for every finite subgrup $H\leq G$. Let $B$ be a $G$-algebra and $\bold{E}$ be the $\Or G$-spectrum defined in \eqref{defe}. Then 
\[H^{G}(E_{\mathcal{F}in} G; \bold{E})\cong \operatornamewithlimits{colim}_{X}\mathbb{K}^{G}(\ell^{(X)},B)\]
where $X$ runs over the $(G,\mathcal{F}in)$-finite subcomplexes of $E_{\mathcal{F}in} G$. 
\end{thm}
The proof of Theorem \ref{thm:mainIntro} has two parts. Let $\OrFin{G}$ be the full subcategory of $\Or{G}$ whose objects are $G/H$ with $H\in\mathcal{F}in$. We consider the $\OrFin{G}$-spectrum 
$$
\bold{F}:\OrFin{G}\rightarrow \mathrm{Sp}\text{,} \qquad \bold{F}(G/H)=\mathbb{K}^{G}(\ell^{(G/H)},B)\text{,}
$$
which provides an homology theory on $(G, \FIN)$-complexes, see details in \cite{dalu}. A \emph{$(G, \FIN)$-complex} is a $G$-simplicial set that is built from cells of the form $G/H\times \Delta^n$ with $H\in\mathcal{F}in$.
The first step in the proof of Theorem \ref{thm:mainIntro} is to prove that for every $(G, \FIN)$-complex $Y$ we obtain:
\begin{equation}\label{parte2}
\framebox{$H^{G}(Y;\bold{E}) \cong H^{G}(Y;\bold{F})$}
\end{equation}
In order to prove the above, 
we construct in Theorem \ref{thm:ZZAG} a zig-zag of $\OrFin{G}$-spectra: 
\[\xymatrix{\bold{E}(G/H)=\mathbb{K}(A, \mathcal{R}(B\rtimes G/H))\ar@{-->}[r] & \mathbb{K}^G(A^{(G/H)}, B)=\bold{F}(G/H)}
\]
The second step is to prove that 
\begin{equation}\label{parte1}
\framebox{$H^{G}(Y;\bold{F}) =\operatornamewithlimits{colim}_{X}\mathbb{K}^{G}(\ell^{(X)},B)$}
\end{equation}
where $Y$ is a $(G, \FIN)$-complex and $X\subseteq Y$ is $(G, \FIN)$-finite. This is proved in Lemma \ref{pruebaparte1}.

The paper is organized as follows. In Section \ref{sec:adtr} we recall the adjointness theorems from \cite{euge}. Upon composing \eqref{gjmor} and \eqref{irmor} we get an isomorphism
\begin{equation}\label{eq:compositeAdj}
    kk^G(A^{(G/H)}, B)\cong kk(A, B\rtimes H)
\end{equation}
for $A\in\mathrm{Alg}$, $B\in{G\mathrm{Alg}_\ell}$ and $H$ a finite subgroup of $G$ such that $\tfrac{1}{|H|}\in\ell$. Along this section we give explicit descriptions of the unit and counit of this adjunction; this is summarized in Proposition \ref{prop:adj}. In Section \ref{sec:crossedpro} we define a triangulated functor $\mathcal{R}(-\rtimes G/H):kk^G\tokk$ that is naturally isomorphic to the crossed product with $H$. This allows us to replace the right-hand side of \eqref{eq:compositeAdj} by $kk(A, \mathcal{R}(B\rtimes G/H))$ and consider it as a covariant functor on $\Or{G}$. In Section \ref{sec:natadj}, we prove that the isomorphism
\begin{equation}\label{eq:compositeAdjNat}
    kk^G(A^{(G/H)}, B)\cong kk(A, \mathcal{R}(B\rtimes G/H))
\end{equation}
is natural in $G/H$; see Theorem \ref{thm:adj}. Moreover, we provide an explicit description for the counit of this adjunction in Lemma \ref{lem:natcounit}. Section \ref{sec:bomba} is the technical core of this work and is devoted to lifting the isomorphism \eqref{eq:compositeAdjNat} to a natural weak equivalence of spectra. By Lemma \ref{lem:natcounit}, we can describe the isomorphism \eqref{eq:compositeAdjNat} as the composite of the morphisms in the zig-zag \eqref{eq:decompAdj}. Upon replacing $kk$ by $\mathbb{K}$ and $kk^G$ by $\mathbb{K}^G$ we obtain a zig-zag:
\begin{equation}\label{eq:zzagpintro}\begin{gathered}\xymatrix@C=3.5em{
\mathbb{K}(A, \mathcal{R}(B\rtimes G/N))\ar[r]^-{(-)^{(G/N)}} & \mathbb{K}^G(A^{(G/N)}, \left[\mathcal{R}(B\rtimes G/N)\right]^{(G/N)})\\
 \mathbb{K}^G(A^{(G/N)}, M_{G}B) & \mathbb{K}^G(A^{(G/N)}, \mathcal{R}\left[(B\rtimes G/N)^{(G/N)}\right])\ar[u]_-{\rotatebox{90}{$\scriptstyle\sim$}}\ar[l]_-{\mathcal{R}(\zeta_{G/N})}}\end{gathered}\end{equation}
Here the technical difficulties arise:
\begin{enumerate}
    \item\label{item:intro1} How to consider the spectra on the right column as covariant functors $\OrFin{G}\to\mathrm{Sp}$ (or replace them by ones)?
    \item\label{item:intro2} Once the previous question has been addressed, are the morphisms in \eqref{eq:zzagpintro} natural in $G/N$?
\end{enumerate}
These issues are explained with greater detail in Section \ref{sec:zzagp}. In Section \ref{sec:notpreli} we introduce a more convenient notation, rewriting \eqref{eq:zzagpintro} as:
\begin{equation}\label{eq:zzagPrimitIntro}\xymatrix{
J(t) \ar[r]^-{(-)^{(t)}} &
M_t(t, t) & L_t(t,t)\ar[l]^-{\sim}_-{\psi}\ar[r]^-{\mathcal{R}(\zeta_t)} &
\mathbb{K}^G(A^{(t)}, M_G B)
}\end{equation}
In Sections \ref{sec:coend} and \ref{sec:coend2} we show how defining $\OrFin{G}$-spectra as objecwise coends provides an answer to \eqref{item:intro1}. In Section \ref{sec:morfi} we define a morphism $\varphi$ of $\OrFin{G}$-spectra that can be thought of as a convenient analogue to the morphism $(-)^{(t)}$ in \eqref{eq:zzagPrimitIntro}. In Section \ref{sec:modcatbi} we introduce a model structure on categories of bifunctors that will allow us to build models for the homotopy coends of certain morphisms of bifunctors. In Section \ref{sec:zznat} we gather the previous results and give an answer to \eqref{item:intro2} in Theorem \ref{thm:ZZAG}. In Section \ref{sec:prin} we finally prove Theorem \ref{thm:mainIntro} and show that the left-hand side of the assembly map for the $KH$-isomorphism conjecture can be expressed in terms of equivariant algebraic $kk$-theory groups.








\section{Adjoint theorems revisited}\label{sec:adtr}
Throughout this text, $\ell$ will denote a commutative ring with unit and $\otimes$ will denote the tensor product over $\ell$. 
We will refer to $\ell$-algebras simply as {\emph{algebras}}. An algebra with an action of a group $G$ will be called a \emph{$G$-algebra}. The following lemma will be useful later on.


\begin{lem}[\cite{friendly}*{Proposition 2.2.6}]\label{lem:conju}
Let $\mathcal{D}$ be a category and let $F:{G\mathrm{Alg}_\ell}\to\mathcal{D}$ be an $M_2$-stable functor. Let $B\subseteq C$ be an inclusion of $G$-algebras and let $V\in C$ be an invertible element such that $VB,BV^{-1}\subseteq B$ and $g\cdot V=V$ for all $g\in G$. Then the formula $\phi^{V}(b)=VbV^{-1}$ defines a $G$-algebra homomorphism $\phi^V:B\to B$ such that $F(\phi^V)=\mathrm{id}_{F(B)}$.
\end{lem}
\begin{proof}
	It is easily verified that $g\cdot V^{-1}=V^{-1}$ for all $g\in G$ and that $\phi^V$ defines a $G$-algebra homomorphism. The rest of the proof of \cite{friendly}*{Proposition 2.2.6} carries over verbatim.
\end{proof}

If $X$ is a set, we will write $M_X$ for the algebra of finite matrices with coefficients in $\ell$ that are indexed over $X\times X$. 
Let $S$ be a $G$-set and let $|S|$ be its underlying set. We will write $M_S$ for the $\ell$-algebra $M_{|S|}$ endowed with the $G$-action defined by $g\cdot e_{s,t}=e_{g\cdot s, g\cdot t}$. For any $B\in{G\mathrm{Alg}_\ell}$, we have an $\ell$-algebra isomorphism $R_{S,B}:(M_S\otimes B)\rtimes G\to M_{|S|}\otimes (B\rtimes G)$ defined by:
\[R_{S,B}((e_{s,t}\otimes b)\rtimes g)=e_{s,g^{-1}t}\otimes (b\rtimes g)\]
This isomorphism is clearly natural in $S$ with respect to injective morphisms of $G$-sets and natural in $B$ with respect to $\ell$-algebra homomorphisms.

Let $G_+=G\coprod\{\ast\}$ and let $\iota:\ell\to M_{G_+}$ (respectively $\iota':M_G\to M_{G_+}$) be the morphism induced by the inclusion $\{\ast\}\subset G$ (resp. $G\subset G_+$). Let $B\in{G\mathrm{Alg}_\ell}$. Recall from \cite{euge}*{(4.1.3)} that we have the following zig-zag of isomorphisms in $kk^G$:
\begin{equation}\label{eq:zigzag}
\xymatrix{B\ar[r]^-{\iota_*}_-{\cong} & M_{G_+}\otimes B & M_G\otimes B\ar[l]_-{(\iota')_*}^-{\cong}}
\end{equation}
Let $G$ be a group and let $H\subseteq G$ be a finite subgroup of order $n$ such that $\frac1n\in\ell$. By \cite{euge}*{Theorem 5.2.1} and \cite{euge}*{Theorem 6.14} we have an adjunction isomorphism
\begin{equation}\label{eq:adj}kk^G(A^{(G/H)}, B)\cong kk(A, B\rtimes H)\end{equation}
for any $A\in{\mathrm{Alg}_\ell}$ and any $B\in{G\mathrm{Alg}_\ell}$. For $A\in{\mathrm{Alg}_\ell}$,  let $\varphi_A:A\to A^{(G/H)}\rtimes H$ be the algebra homomorphism  defined by:
\[\varphi_A(a)=a\chi_H\rtimes \frac1n\sum_{h\in H}h\]
Put $\eta_A=j(\varphi_A)\in kk(A,A^{(G/H)}\rtimes H)$. We will show that $\eta_A$ is a unit for the adjunction \eqref{eq:adj}. For $B\in{G\mathrm{Alg}_\ell}$, let $\psi_B:(B\rtimes H)^{(G/H)}\to M_G\otimes B$ be the $G$-algebra homomorphism defined by:
\begin{equation}\label{eq:defipsi}\psi_B\left((b\rtimes h)\chi_{wH}\right)=\sum_{p\in wH}e_{p,ph}\otimes p(b)\end{equation}
Let $\varepsilon_B\in kk^G((B\rtimes H)^{(G/H)},B)$ be the following composite in $kk^G$, where the isomorphism on the right is given by the zig-zag \eqref{eq:zigzag}:
\begin{equation}\label{eq:couniH}\xymatrix{(B\rtimes H)^{(G/H)}\ar[r]^-{j^G(\psi_B)} & M_G\otimes B\cong B}\end{equation}
We will show that $\varepsilon_B$ is a counit for the adjunction \eqref{eq:adj}.

\begin{lem}\label{lem:unit}
For any $B\in{G\mathrm{Alg}_\ell}$ we have $(\varepsilon_B\rtimes H)\circ \eta_{B\rtimes H}=\mathrm{id}_{B\rtimes H}$ in $kk$.
\end{lem}
\begin{proof}
It is easily verified that the following diagram in $kk$ commutes, where all the arrows are isomorphisms:
\[\xymatrix{(M_G\otimes B)\rtimes H\ar[d]_-{(R_{G,B})_*}\ar[r]^-{(\iota')_*} & (M_{G_+}\otimes B)\rtimes H\ar[d]_-{(R_{G_+,B})_*} & B\rtimes H\ar[l]_-{\iota_*}\ar@/^1pc/[dl]^-{\iota_*} \\
M_{|G|}\otimes (B\rtimes H)\ar[r]_-{(\iota')_*} & M_{|G_+|}\otimes (B\rtimes H)}\]
Thus, the isomorphism $(M_G\otimes B)\rtimes H\cong B\rtimes H$ in $kk$ induced by the zig-zag \eqref{eq:zigzag} equals the composite:
\[\xymatrixcolsep{4em}\xymatrix{(M_G\otimes B)\rtimes H\ar[r]^-{j(R_{G,B})} & M_{|G|}\otimes (B\rtimes H) & B\rtimes H\ar[l]_-{j(e_{1,1}\otimes ?)}}\]
To prove the lemma, it will be enough to show that the composite
\[\xymatrix{B\rtimes H \ar[r]^-{\varphi_{B\rtimes H}} & (B\rtimes H)^{(G/H)}\rtimes H\ar[r]^-{\psi_B\rtimes H} & (M_G\otimes B)\rtimes H\ar[r]^-{R_{G,B}} & M_{|G|}\otimes (B\rtimes H)}\]
 and the inclusion $e_{1,1}\otimes ?:B\rtimes H\to M_{|G|}\otimes (B\rtimes H)$ induce the same morphism in $kk$. Let $\Gamma$ be the algebra of matrices with coefficients in $\tilde{B}\rtimes H$ indexed by $G\times G$ that have only finitely many nonzero coefficients in each column and each row. Notice that $\Gamma$ is a unital algebra that contains $M_{|G|}\otimes (B\rtimes H)$ as a subalgebra. Let $V=\sum_{g\in G}e_{g,g}\otimes (1\rtimes g)\in\Gamma$. We have:
\begin{align*}[R_{G,B}\circ (\psi_B\rtimes H)\circ \varphi_{B\rtimes H}](b\rtimes h)&=\frac1n\sum_{p,q\in H}e_{p,q}\otimes (p(b)\rtimes phq^{-1}) \\
&=V\left(\frac1n\sum_{p,q\in H}e_{p,q}\otimes (b\rtimes h)\right)V^{-1}\end{align*}
Moreover, $\frac1n\sum_{p,q\in H}e_{p,q}$ is a conjugate of $e_{1,1}$ in $M_{|G|}$; see \cite{euge}*{Remark 3.1.11}. By \cite{friendly}*{Proposition 2.2.6}, we have
\[j[R_{G,B}\circ (\psi_B\rtimes H)\circ \varphi_{B\rtimes H}]=j(e_{1,1}\otimes ?):B\rtimes H\to M_{|G|}\otimes (B\rtimes H),\]
as we wanted to prove.
\end{proof}

\begin{lem}\label{lem:counit}
	For any $A\in{\mathrm{Alg}_\ell}$ we have $\varepsilon_{A^{(G/H)}}\circ\left[\left(\eta_A\right)^{(G/H)}\right]=\mathrm{id}_{A^{(G/H)}}$ in $kk^G$.
\end{lem}
\begin{proof}
	It is easily verified that the composite
	\[\xymatrixcolsep{3em}\xymatrix{A^{(G/H)}\ar[r]^-{(\eta_A)^{(G/H)}} & \left(A^{(G/H)}\rtimes H\right)^{(G/H)}\ar[r]^-{\varepsilon_{A^{(G/H)}}} & M_G\otimes A^{(G/H)}\ar[r]^-{\iota'} & M_{G_+}\otimes A^{(G/H)}}\]
	is given by:
	\begin{equation}\label{eq:compo}a\chi_{wH}\mapsto \frac1n\sum_{p,q\in H}e_{wp,wq}\otimes a\chi_{wH} \end{equation}
	To prove the lemma, it suffices to show that the above formula induces the same morphism as $\iota:A^{(G/H)}\to M_{G_+}\otimes A^{(G/H)}$, $\iota(a\chi_{wH})=e_{*,*}\otimes a\chi_{wH}$, upon applying $j^G:{G\mathrm{Alg}_\ell}\to kk^G$. Let $\tilde{A}$ be the unitalization of $A$ and let $\Gamma_{G_+}(\tilde{A})$ be the set of those matrices with coefficients in $\tilde{A}$, indexed over $G_+\times G_+$, that have finitely many nozero coefficients in each row and each column. Then $\Gamma_{G_+}(\tilde{A})$ is a $G$-algebra with the usual matrix multiplication and the $G$-action defined by $(g\cdot a)_{x,y}:=a_{g^{-1}\cdot x, g^{-1}\cdot y}$. Moreover, we have inclusions of $G$-$\ell$-algebras as follows:
	\[M_{G_+}\otimes A^{(G/H)}\subseteq \left(M_{G_+}\otimes A\right)^{G/H}\subseteq \left(\Gamma_{G_+}(\tilde{A})\right)^{G/H}\]
	The $G$-action on the right is described by
	\[(g\cdot f)(tH):=g\cdot f(g^{-1}tH),\]
	where $f:G/H\to \Gamma_{G_+}(\tilde{A})$ is a function and $g\in G$. We will show that there exists an invertible $V\in(\Gamma_{G_+}(\tilde{A}))^{G/H}$ such that the following diagram commutes, where the horizontal morphism is given by \eqref{eq:compo}:
	\begin{equation}\label{eq:phiV}\begin{gathered}\xymatrix{A^{(G/H)}\ar[r]\ar@/_/[dr]_-{\iota} & M_{G_+}\otimes A^{(G/H)}\ar[d]^-{\phi^V} \\
	& M_{G_+}\otimes A^{(G/H)}}\end{gathered}\end{equation}
	Once this is done, the result will follow by Lemma \ref{lem:conju} since $j^G:{G\mathrm{Alg}_\ell}\to kk^G$ is $M_2$-stable. For $wH\in G/H$, define:
	\[V_{wH}:=\sum_{x\in wH}\left(\tfrac{n-1}{n}e_{x,x}+e_{*,x}+e_{x,*}\right)-\sum_{x,y\in wH}\tfrac{1}{n}e_{x,y}+\sum_{g\in G\setminus wH}e_{g,g}\in \Gamma_{G_+}(\tilde{A})\]
	Note that $g\cdot V_{wH}=V_{gwH}$ for all $g\in G$. We can picture $V_{wH}$ as a block diagonal matrix having an identity block in the coordinates corresponding to $g\in G\setminus wH$, and the following block in the coordinates corresponding to elements of $wH\cup\{\ast\}$:
	 \[
	\begin{tikzpicture}[mymatrixenv]
	\matrix[mymatrix] (m)  {
		\frac{n-1}{n}  & -\frac{1}{n} &
		\cdots & -\frac{1}{n} & -\frac{1}{n} &[0.5em,between borders] 1 \\
		-\frac{1}{n} & \frac{n-1}{n} &
		\cdots & -\frac{1}{n} & -\frac{1}{n} & 1 \\
		%
		\vdots & & \ddots & & \vdots & \vdots \\
		%
		-\frac{1}{n} & -\frac{1}{n} &
		\cdots & \frac{n-1}{n} & -\frac{1}{n} & 1 \\
		%
		-\frac{1}{n} & -\frac{1}{n} &
		\cdots & -\frac{1}{n} & \frac{n-1}{n} & 1 \\[0.7em]
		%
		1 & 1 & \cdots & 1 & 1 & 0 \\
	};
	\draw[dashed]([xshift=0.8em]m.north west -| m-1-5.east) to ([xshift=0.8em] m.south east -| m-2-5.east);
	\draw[dashed]([xshift=-0.4em,yshift=0.7em)]m-6-1.north west) to ([yshift=0.7em,xshift=0.4em] m-6-6.north east); 
	\mymatrixbraceright{1}{5}{$wH$}
	\mymatrixbracetop{1}{5}{$wH$}
	\end{tikzpicture}
	\]
	It is easily verified that $V_{gH}$ is invertible with inverse given by:
	\[V_{wH}^{-1}:=\sum_{x\in wH}\left(\tfrac{n-1}{n}e_{x,x}+\tfrac1ne_{*,x}+\tfrac1n e_{x,*}\right)-\sum_{x,y\in wH}\tfrac{1}{n}e_{x,y}+\sum_{g\in G\setminus wH}e_{g,g}\in \Gamma_{G_+}(\tilde{A})\]
	Again, we can think of $V_{wH}^{-1}$ as a block diagonal matrix having an identity block in the coordinates corresponding to $g\in G\setminus wH$, and the following block in the coordinates corresponding to elements of $wH\cup\{\ast\}$:
	\[
	\begin{tikzpicture}[mymatrixenv]
	\matrix[mymatrix] (m)  {
		\frac{n-1}{n}  & -\frac{1}{n} &
		\cdots & -\frac{1}{n} & -\frac{1}{n} &[0.5em,between borders] \frac1n \\
		-\frac{1}{n} & \frac{n-1}{n} &
		\cdots & -\frac{1}{n} & -\frac{1}{n} & \frac1n \\
		%
		\vdots & & \ddots & & \vdots & \vdots \\
		%
		-\frac{1}{n} & -\frac{1}{n} &
		\cdots & \frac{n-1}{n} & -\frac{1}{n} & \frac1n \\
		%
		-\frac{1}{n} & -\frac{1}{n} &
		\cdots & -\frac{1}{n} & \frac{n-1}{n} & \frac1n \\[0.7em]
		%
		\frac1n & \frac1n & \cdots & \frac1n & \frac1n & 0 \\
	};
	\draw[dashed]([xshift=0.8em]m.north west -| m-1-5.east) to ([xshift=0.8em] m.south east -| m-2-5.east);
	\draw[dashed]([xshift=-0.4em,yshift=0.7em)]m-6-1.north west) to ([yshift=0.7em,xshift=0.4em] m-6-6.north east); 
	\mymatrixbraceright{1}{5}{$wH$}
	\mymatrixbracetop{1}{5}{$wH$}
	\end{tikzpicture}
	\]
	Define $V^{\pm 1}:G/H\to \Gamma_{G_+}(\tilde{A})$ by $V^{\pm 1}(wH):=V_{wH}^{\pm 1}$. Then $V,V^{-1}\in (\Gamma_{G_+}(\tilde{A}))^{G/H}$ are mutual inverses and
	\[V(M_{G_+}\otimes A^{(G/H)}),(M_{G_+}\otimes A^{(G/H)})V^{-1}\subseteq (M_{G_+}\otimes A^{(G/H)}).\]
	Moreover, $g\cdot V=V$ for all $g\in G$.  An easy computation shows that the triangle \eqref{eq:phiV} commutes. This finishes the proof.
\end{proof}

\begin{prop}[cf. \cite{euge}*{Theorems 5.2.1 and 6.14}]\label{prop:adj}
	Let $G$ be a group and let $H\subseteq G$ be a finite subgroup of order $n$ such that $\frac1n\in\ell$. Then the morphisms $\eta_A\in kk(A,A^{(G/H)}\rtimes H)$ and $\varepsilon_B\in kk^G((B\rtimes H)^{(G/H)},B)$ defined above are respectively the unit and the counit of an adjunction:
	\begin{equation}\label{eq:adjnnat}
		kk^G(A^{(G/H)}, B)\cong kk(A, B\rtimes H)
	\end{equation}
\end{prop}
\begin{proof}
	It follows immediately from Lemmas \ref{lem:unit} and \ref{lem:counit}.
\end{proof}


\section{Crossed product with \texorpdfstring{$G/H$}{G//H}}\label{sec:crossedpro}

We want to show that the adjunction \eqref{eq:adjnnat} is natural in $G/H$. The first thing to do is to replace the right-hand side by an actual functor on $\OrFin G$.

\subsection{Non unital \texorpdfstring{$\ell$}{l}-linear categories}\label{sec:lincat}

\begin{defi}
A \emph{non-unital $\ell$-linear category} $\mathcal{C}$ consists of:
\begin{enumerate}
	\item a set of objects ${\mathrm{ob}\,}\mathcal{C}$,
	\item an $\ell$-module $\mathcal{C}(x,y)$ for every $x,y\in{\mathrm{ob}\,}\mathcal{C}$, and
	\item $\ell$-module homomorphisms
	\begin{equation}\label{eq:compC} \circ:\mathcal{C}(y,z)\otimes\mathcal{C}(x,y)\to\mathcal{C}(x,z) \end{equation}
	for every $x,y,z\in{\mathrm{ob}\,}\mathcal{C}$, that are associative in the obvious way.
\end{enumerate}
\end{defi}

Non-unital $\ell$-linear categories with only one object can be identified with (non-unital) $\ell$-algebras. In the sequel, we will refer to non-unital $\ell$-linear categories simply as \emph{$\ell$-linear categories}.

\begin{defi}
Let $\mathcal{C}$ and $\mathcal{D}$ be $\ell$-linear categories. An $\ell$-linear functor $F:\mathcal{C}\to\mathcal{D}$ consists of a function $F:{\mathrm{ob}\,}\mathcal{C}\to{\mathrm{ob}\,}\mathcal{D}$ together with $\ell$-module homorphisms
\begin{equation}\label{eq:linearF} F_{x,y}:\mathcal{C}(x,y)\to\mathcal{D}(F(x),F(y)) \end{equation}
that are compatible with the composition.
\end{defi}

We will write ${\mathrm{Cat}_\ell}$ for the category whose objects are $\ell$-linear categories and whose morphisms are $\ell$-linear functors.

Let $\mathcal{C}$ and $\mathcal{D}$ be $\ell$-linear categories. The \emph{tensor product} $\mathcal{C}\otimes \mathcal{D}$ is the $\ell$-linear category with objects ${\mathrm{ob}\,}(\mathcal{C})\otimes{\mathrm{ob}\,}(\mathcal{D})$ and such that:
\[(\mathcal{C}\otimes\mathcal{D})((c,d),(\tilde{c}, \tilde{d}))=\mathcal{C}(c,\tilde{c})\otimes\mathcal{D}(d, \tilde{d})\] Composition is defined in the usual way, using the composition laws in $\mathcal{C}$ and $\mathcal{D}$ and the commutativity of the tensor product of $\ell$-modules.

We proceed to recall some definitions from \cite{corel}*{Section 3}. Let $\mathcal{C}$ be an $\ell$-linear category. Put:
\[
\mathcal{A}(\mathcal{C})=\bigoplus_{x,y\in{\mathrm{ob}\,} \mathcal{C}}\mathcal{C}(x,y)
\]
If $f\in\mathcal{A}(\mathcal{C})$, write $f_{y,x}$ for its component in $\mathcal{C}(x,y)$. Then $\mathcal{A}(\mathcal{C})$ is an $\ell$-algebra with multiplication given by
\[
(gf)_{y,x}=\sum_{z\in{\mathrm{ob}\,}\mathcal{C}}g_{y,z}\circ f_{z,x}
\]

\begin{exa}
Let $\mathcal{C}$ and $\mathcal{D}$ be $\ell$-linear categories. It is easily verified that:
\[\mathcal{A}(\mathcal{C}\otimes\mathcal{D})\cong\mathcal{A}(\mathcal{C})\otimes\mathcal{A}(\mathcal{D})\]
\end{exa}

\begin{exa}\label{exa:subalgebraMGH}
Let $A\in{\mathrm{Alg}_\ell}$, let $G$ be a group and let $H$ be a subgroup. We can regard $\mathcal{A}\left(A\rtimes\GG{G}{G/H}\right)$ as a subalgebra of $M_{|G/H|}(A\rtimes G)$ using the inclusion that sends $a\rtimes g\in\mathrm{Hom}_{A\rtimes\GG{G}{G/H}}(uH, vH)$ to $e_{vH,uH}\cdot (a\rtimes g)$.
\end{exa}

A problem with $\mathcal{A}(\mathcal{C})$ is that it is not natural with respect to all $\ell$-linear functors, but only with respect to those that are injective on objects; see \cite{corel}*{p.1231}. To fix this, one defines the $\ell$-algebra $\mathcal{R}(\mathcal{C})$ \cite{corel}*{Section 3.4}. If $M$ is an $\ell$-module, write $T(M)=\oplus_{n\geq 1}M^{\otimes n}$ for the unaugmented tensor algebra. Put:
\[
\mathcal{R}(\mathcal{C})=T\left(\mathcal{A}(\mathcal{C})\right)/\langle\left\{g\otimes f-g\circ f: f\in\mathcal{C}(x,y),\, g\in\mathcal{C}(y,z),\, x,y,z\in{\mathrm{ob}\,}\mathcal{C}\right\}\rangle
\]
This defines a functor $\mathcal{R}:{\mathrm{Cat}_\ell}\to{\mathrm{Alg}_\ell}$.

\begin{rem}
Let $G$ be a group. One can define a $G$-category as an $\ell$-linear category $\mathcal{C}$ such that the hom-modules $\mathcal{C}(x,y)$ are $G$-modules and the composition law \eqref{eq:compC} is $G$-equivariant, endowing $\mathcal{C}(y,z)\otimes\mathcal{C}(x,y)$ with the diagonal $G$-action. If $\mathcal{C}$ and $\mathcal{D}$ are $G$-categories, a $G$-functor $F:\mathcal{C}\to\mathcal{D}$ is an $\ell$-linear functor such that the morphisms \eqref{eq:linearF} are $G$-equivariant. These definitions give rise to a category ${G\mathrm{Cat}_\ell}$ whose objects are $G$-categories and whose morphisms are $G$-functors.

If $\mathcal{C}$ is a $G$-category, then $\mathcal{A}(\mathcal{C})$ and $\mathcal{R}(\mathcal{C})$ are $G$-algebras in a natural way. Thus, we have a functor $\mathcal{R}:{G\mathrm{Cat}_\ell}\to{G\mathrm{Alg}_\ell}$.
\end{rem}

\begin{exa}\label{exa:rtensor}
Let $\mathcal{C}$ be an $\ell$-linear category and $D\in{\mathrm{Alg}_\ell}$. We claim that there is a natural morphism:
\[\mathcal{R}(\mathcal{C}\otimes D)\rightarrow \mathcal{R}(\mathcal{C})\otimes D\]
To see this, first note that there is an $\ell$-linear functor $\mathcal{C}\to\mathcal{R}(\mathcal{C})$ that takes $f\in\mathcal{C}(x,y)$ to the class in $\mathcal{R}(\mathcal{C})$ of $f\in\mathcal{C}(x,y)\subseteq \mathcal{A}(\mathcal{C})\subseteq T(\mathcal{A}(\mathcal{C}))$. Upon tensoring this functor with $D$ and then applying $\mathcal{R}(-)$, we get the desired morphism:
\begin{equation}\label{eq:Rtensor}\mathcal{R}(\mathcal{C}\otimes D)\to \mathcal{R}(\mathcal{R}(\mathcal{C})\otimes D)=\mathcal{R}(\mathcal{C})\otimes D\end{equation}
If $\mathcal{C}$ is a $G$-category and $D$ is a $G$-algebra, then \eqref{exa:rtensor} is a morphism of $G$-algebras.
\end{exa}

Is $\mathcal{C}$ is an $\ell$-linear category, there is a morphism $p:\mathcal{R}(\mathcal{C})\to\mathcal{A}(\mathcal{C})$ induced by multiplication in $\mathcal{A}(\mathcal{C})$.

\begin{lem}[cf. \cite{corel}*{Lemma 3.4.3}]\label{lem:piso} Let $\mathcal{C}$ be an $\ell$-linear category (respectively a $G$-category). Then the morphism
\[p:\mathcal{R}(\mathcal{C})\to\mathcal{A}(\mathcal{C})\]
induces an isomorphism in $kk$ (resp. in $kk^G$).
\end{lem}
\begin{proof}
    The proof of \cite{corel}*{Lemma 3.4.3} carries on verbatim in this setting.
\end{proof}

\begin{coro}\label{coro:rtensor}
Let $\mathcal{C}\in{\mathrm{Cat}_\ell}$ and $D\in{\mathrm{Alg}_\ell}$ (resp. $\mathcal{C}\in{G\mathrm{Cat}_\ell}$ and $D\in{G\mathrm{Alg}_\ell}$). Then the morphism
\[\mathcal{R}(\mathcal{C}\otimes D)\rightarrow \mathcal{R}(\mathcal{C})\otimes D\]
of Example \ref{exa:rtensor} is an isomorphism in $kk$ (resp. in $kk^G$).
\end{coro}
\begin{proof}
    It is easily verified that the following diagram commutes in  ${\mathrm{Alg}_\ell}$ (resp. in ${G\mathrm{Alg}_\ell}$):
    \[\xymatrix{\mathcal{R}(\mathcal{C}\otimes D)\ar[r]\ar[d]_-{p} & \mathcal{R}(\mathcal{C})\otimes D\ar[d]^-{p\otimes \mathrm{id}_{D}} \\
    \mathcal{A}(\mathcal{C}\otimes D)\ar[r]^-{\cong} & \mathcal{A}(\mathcal{C})\otimes D}\]
    Indeed, it suffices to check commutativity on the generators $f\otimes d$, and this is immediate. The result follows from Lemma \ref{lem:piso}.
\end{proof}


\subsection{Crossed product with \texorpdfstring{$G/H$}{G/H}}\label{sec:crospro}
Fix $G/H\in\Or G$ and write $-\rtimes G/H$ for the functor $-\rtimes \GG{G}{G/H}:{G\mathrm{Alg}_\ell}\to {\mathrm{Cat}_\ell}$. We claim that the composite functor
\begin{equation}\label{eq:compFunc}
\xymatrix@C=6em{{G\mathrm{Alg}_\ell} \ar[r]^-{-\rtimes G/H} & {\mathrm{Cat}_\ell}\ar[r]^-{\mathcal{R}} & {\mathrm{Alg}_\ell}\ar[r]^-{j} & kk}
\end{equation}
factors through $j^G:{G\mathrm{Alg}_\ell}\tokk^G$. To prove this, it suffices to show that it is excisive, homotopy invariant and $G$-stable \cite{euge}*{Theorem 4.1.1}. Homotopy invariance and $G$-stability follow easily from the following.

\begin{lem}\label{lem:natisocompoGH}
Let $H$ be a subgroup of $G$ and let $uH\in G/H$. Write $\pow{u}{H}$ for the conjugate $uHu^{-1}$.
Then the composite functor in \eqref{eq:compFunc} is naturally isomorphic to:
\begin{equation}\label{eq:jrtimesH}
A\mapsto j({\mathrm{Res}}_G^{\pow{u}{H}}(A)\rtimes \pow{u}{H})\text{.}\end{equation}
\end{lem}
\begin{proof}
    To ease notation, we will omit ${\mathrm{Res}}_G^{\pow{u}{H}}$ throughout the proof and write $A\rtimes \pow{u}{H}$ instead of ${\mathrm{Res}}_G^{\pow{u}{H}}(A)\rtimes \pow{u}{H}$. Let $B\in{G\mathrm{Alg}_\ell}$. Consider $B\rtimes \pow{u}{H}\subseteq B\rtimes G/H$ as the full subcategory whose only object is $uH$. Upon applying $\mathcal{R}$ to this inclusion, we get an algebra homomorphism:
    \[B\rtimes \pow{u}{H}=\mathcal{R}(B\rtimes \pow{u}{H})\longrightarrow \mathcal{R}\left(B\rtimes G/H\right)\]
    Let $\nu_{uH}$ be the image of this morphism in $kk$. Clearly, $\nu_{uH}$ is a natural transformation from \eqref{eq:jrtimesH} to the composite \eqref{eq:compFunc}. To finish the proof, we will show that $\nu_{uH}$ is an isomorphism. We claim that there is an isomorphism $\alpha$ that fits in the following commutative diagram in $kk$:
    \[\xymatrix@C=3em{B\rtimes \pow{u}{H}\ar[r]^-{\nu_{uH}}\ar@/_3pc/[ddr]_-{\cdot e_{uH,uH}} & \mathcal{R}\left(B\rtimes G/H\right)\ar[d]^-{p}_-{\simeq} \\
    & \mathcal{A}\left(B\rtimes G/H\right)\ar[d]_-{\simeq}^-{\alpha} \\ 
    & M_{|G/H|}(B\rtimes \pow{u}{H}) }\]
    Here, the bent arrow is induced by the inclusion into the $(uH,uH)$-coefficient and it is an isomorphism by matrix invariance. It follows that $\nu_{uH}$ is an isomorphism too. The isomorphism $\alpha$ is constructed as in the proof of \cite{corel}*{Lemma 3.2.6}. More precisely, let $s:G/H\to G$ be a section of the projection such that $s(uH)=u$. Write $\hat{g}=s(gH)$ for $g\in G$. For $b\in B$ and $g\in\hom_{\GG{G}{G/H}}(sH,tH)$ put:
    \[\alpha(b\rtimes g)=e_{tH,sH}\cdot (u\hat{t}^{-1}(b)\rtimes u\hat{t}^{-1}g\hat{s}u^{-1})\]
    It is straightforward to verify that this formula defines an isomorphism of algebras $\alpha:\mathcal{A}\left(B\rtimes G/H\right)\to M_{|G/H|}(B\rtimes \pow{u}{H})$.
\end{proof}

\begin{coro}[cf. \cite{euge}*{Proposition 5.1.2 and Section 6}]\label{coro:rtimesGHhtpyinv}
The composite functor \eqref{eq:compFunc} is homotopy invariant and $G$-stable.
\end{coro}
\begin{proof}
    Write $F:{G\mathrm{Alg}_\ell}\tokk$ for the functor $F=j({\mathrm{Res}}_G^H(-)\rtimes H)$. By Lemma \ref{lem:natisocompoGH} it suffices to show that $F$ is homotopy invariant and $G$-stable. By definition, $F$ is the composite:
    \[\xymatrix@C=4em{{G\mathrm{Alg}_\ell}\ar[r]^-{{\mathrm{Res}}_G^H} & {H\mathrm{Alg}_\ell}\ar[r]^-{j(-\rtimes H)} & kk}\]
    The functor $j(-\rtimes H)$ is homotopy invariant and $H$-stable by \cite{euge}*{Proposition 5.1.2}. The functor ${\mathrm{Res}}_G^H$ is easily seen to send homotopic morphisms in ${G\mathrm{Alg}_\ell}$ to homotopic morphisms in ${H\mathrm{Alg}_\ell}$. It follows that $F$ is homotopy invariant. Recall the definition of $G$-stable functor from \cite{euge}*{Section 3.1}. Let $(\mathcal{W}_1, B_1)$ and $(\mathcal{W}_2, B_2)$ be $G$-modules by locally finite automorphisms such that $\mathrm{card}(B_i)\leq \mathrm{card}(\mathbb{N})$ for $i=1,2$ and let $A$ be a $G$-algebra. Then $(\mathcal{W}_1, B_1)$ and $(\mathcal{W}_2, B_2)$ are $H$-modules by locally finite automorphisms and thus
    \[{\mathrm{Res}}_G^H(\mathrm{End}_{\ell}^F(\mathcal{W}_1))\otimes{\mathrm{Res}}_G^HA\to{\mathrm{Res}}_G^H(\mathrm{End}_{\ell}^F(\mathcal{W}_1\oplus \mathcal{W}_2))\otimes{\mathrm{Res}}_G^HA\]
    \[f\mapsto\begin{pmatrix}f & 0 \\ 0 & 0\end{pmatrix}\]
    becomes an isomorphism upon applying $j(-\rtimes H)$. It follows that $F$ is $G$-stable.
    \end{proof}

We still have to show that the composite \eqref{eq:compFunc} is excisive.

\begin{cons}\label{cons:partial}
Let $\mathcal{E}:A\to B\to C$ be an extension in ${G\mathrm{Alg}_\ell}$ (that splits in ${G\mathrm{Mod}_\ell}$). We will to construct a triangle in $kk$ as follows, that is natural with respect to morphisms of extensions:
\[\xymatrix@C=2.7em{\Omega(\mathcal{R}(C\rtimes G/H))\ar[r]^-{\partial_{\mathcal{E}\rtimes G/H}} & \mathcal{R}(A\rtimes G/H)\ar[r] & \mathcal{R}(B\rtimes G/H)\ar[r] & \mathcal{R}(C\rtimes G/H)}\]
Let $\nu_H:j({\mathrm{Res}}_G^H(-)\rtimes H)\to j\mathcal{R}(-\rtimes G/H)$ be the natural isomorphism of Lemma \ref{lem:natisocompoGH}. Note that $j({\mathrm{Res}}_G^H(-)\rtimes H)$ is excisive, since ${\mathrm{Res}}_G^H$ sends extensions in ${G\mathrm{Alg}_\ell}$ to extensions in ${H\mathrm{Alg}_\ell}$ and $j(-\rtimes H):{H\mathrm{Alg}_\ell}\tokk$ is excisive \cite{euge}*{Proposition 5.1.2}. To simplify notation, we omit explicit mention to $j$ and ${\mathrm{Res}}_G^H$. For example, we write $A\rtimes H$ instead of $j({\mathrm{Res}}_G^H(A)\rtimes H)$. We have the following commutative diagram of solid arrows in $kk$, where the top row is a triangle:
\[\xymatrix@C=2.7em{\Omega(C\rtimes H)\ar[r]^-{\partial_{\mathcal{E}\rtimes H}}\ar[d]_-{\simeq}^-{\Omega(\nu_H)} & A\rtimes H\ar[d]_-{\simeq}^-{\nu_H}\ar[r] & B\rtimes H\ar[d]_-{\simeq}^-{\nu_H}\ar[r] & C\rtimes H\ar[d]_-{\simeq}^-{\nu_H} \\
\Omega(\mathcal{R}(C\rtimes G/H))\ar@{-->}[r] & \mathcal{R}(A\rtimes G/H)\ar[r] & \mathcal{R}(B\rtimes G/H)\ar[r] & \mathcal{R}(C\rtimes G/H)}\]
Define $\partial_{\mathcal{E}\rtimes G/H}:\Omega(\mathcal{R}(C\rtimes G/H))\to \mathcal{R}(A\rtimes G/H)$ to be the dashed arrow that makes the left square commute. Then the bottom row becomes a triangle too. This triangle is clearly natural with respect to the extension $\mathcal{E}$. These morphisms $\partial_{\mathcal{E}\rtimes G/H}$ make the composite \eqref{eq:compFunc} into an excisive homology theory.
\end{cons}

\begin{prop}\label{prop:existencertimesGH}
Let $H$ be subgroup of $G$. Then there exists a unique triangulated functor $\overline{-\rtimes G/H}:kk^G\tokk$ that makes the following diagram commute:
\[\xymatrix@C=6em{{G\mathrm{Alg}_\ell}\ar[d]_-{j^G}\ar[r]^-{\mathcal{R}(-\rtimes G/H)} & {\mathrm{Alg}_\ell}\ar[d]^-{j} \\
kk^G\ar[r]_-{\overline{-\rtimes G/H}} & kk}\]
Moreover, for every extension $\mathcal{E}:A\to B\to C$ in ${G\mathrm{Alg}_\ell}$ and every $uH\in G/H$, the following square in $kk$ commutes
\begin{equation}\label{eq:squarepartial}\begin{gathered}\xymatrix{\Omega(C\rtimes \pow{u}{H})\ar[d]_-{\Omega(\nu_{uH})}^-{\simeq}\ar[r]^-{\partial_{\mathcal{E} \rtimes \pow{u}{H}}} & A\rtimes\pow{u}{H}\ar[d]^-{\nu_{uH}}_-{\simeq} \\
\Omega(\mathcal{R}(C\rtimes G/H))\ar[r]^-{\partial_{\mathcal{E}\rtimes G/H}} & \mathcal{R}(A\rtimes G/H)}\end{gathered}\end{equation}
\end{prop}
\begin{proof}
    The functor composite functor \eqref{eq:compFunc} is homotopy invariant and $G$-stable by Corollary \ref{coro:rtimesGHhtpyinv}. Moreover, endowed with the morphisms $\partial_{\mathcal{E}\rtimes G/H}$ defined in Construction \ref{cons:partial}, it becomes an excisive homology theory. Then the existence of $\overline{-\rtimes G/H}:kk^G\tokk$ follows from \cite{euge}*{Theorem 4.1.1}.
    
    To prove the assertion about \eqref{eq:squarepartial}, consider the following diagram in $kk$:
    \begin{equation}\label{eq:partialuH}\begin{gathered}
    \xymatrix{\Omega(C\rtimes \pow{u}{H})\ar[d]_-{\Omega(\nu_{uH})}^-{\simeq}\ar[r]^-{\partial_{\mathcal{E} \rtimes \pow{u}{H}}} & A\rtimes\pow{u}{H}\ar[d]^-{\nu_{uH}}_-{\simeq} \\
\Omega(\mathcal{R}(C\rtimes G/H))\ar[r]^-{\partial_{\mathcal{E}\rtimes G/H}}\ar[d]_-{\Omega(\nu_H)^{-1}}^-{\simeq} & \mathcal{R}(A\rtimes G/H)\ar[d]^-{(\nu_H)^{-1}}_-{\simeq} \\
\Omega(C\rtimes H)\ar[r]^-{\partial_{\mathcal{E}\rtimes H}} & A\rtimes H}
    \end{gathered}\end{equation}
    The bottom square commutes by definition of $\partial_{\mathcal{E}\rtimes G/H}$; see Construction \ref{cons:partial}. Thus, the commutativity of \eqref{eq:squarepartial} is equivalent to that of the outer square in \eqref{eq:partialuH}. We will see that the latter commutes since it is induced by a morphism of extensions in ${\mathrm{Alg}_\ell}$. For $D\in{G\mathrm{Alg}_\ell}$, let $\varphi_u:D\rtimes\pow{u}{H}\to D\rtimes H$ be the algebra homomorphism defined by $\varphi_u(a\rtimes g)=u^{-1}(a)\rtimes u^{-1}gu$. We have a morphism of extensions
    \[
    \xymatrix{A\rtimes\pow{u}{H}\ar[d]^-{\varphi_u}\ar[r] & B\rtimes\pow{u}{H}\ar[d]^-{\varphi_u}\ar[r] & C\rtimes\pow{u}{H}\ar[d]^-{\varphi_u} \\
    A\rtimes H\ar[r] & B\rtimes H\ar[r] & C\rtimes H}
    \]
    that induces a morphism of triangles in $kk$. In particular, there is a commutative square in $kk$ as follows:
    \[\xymatrix@C=4em{\Omega(C\rtimes\pow{u}{H})\ar[d]_-{\Omega j(\varphi_u)}\ar[r]^-{\partial_{\mathcal{E}\rtimes \pow{u}{H}}} & A\rtimes\pow{u}{H}\ar[d]^-{j(\varphi_u)} \\
    \Omega(C\rtimes H)\ar[r]^-{\partial_{\mathcal{E}\rtimes H}} & A\rtimes H}\]
    This square turns out to be the outer square of \eqref{eq:partialuH}. To see this, it suffices to show that $j(\varphi_u)=(\nu_H)^{-1}\circ \nu_{uH}$ or, equivalently, that
    \begin{equation}\label{eq:nuHnuuH}j(p)\circ\nu_H\circ j(\varphi_u)=j(p)\circ\nu_{uH}
    \end{equation}
    where $p:\mathcal{R}\left(D\rtimes G/H\right)\longrightarrow\mathcal{A}\left(D\rtimes G/H\right)$. Each side of \eqref{eq:nuHnuuH} is induced by an algebra homomorphism $D\rtimes \pow{u}{H}\to \mathcal{A}\left(D\rtimes G/H\right)$. We will show that both morphisms are conjugate in $M_{|G/H|}(D\rtimes G)$---regarding $\mathcal{A}\left(D\rtimes G/H\right)$ as a subalgebra of $M_{|G/H|}(D\rtimes G)$ with the inclusion of Example \ref{exa:subalgebraMGH}. A straightforward verification shows that the left- and the right-hand sides of \eqref{eq:nuHnuuH} are induced, respectively, by the algebra homomorphisms $\lambda$ and $\rho$ defined by:
    \begin{align*}
    \lambda(d\rtimes g)&=e_{H,H}\cdot (u^{-1}(d)\rtimes u^{-1}gu) \\
    \rho(d\rtimes g)&=e_{uH, uH}\cdot (d\rtimes g)
    \end{align*}
    Now put:
    \[V=\sum_{vH\in G/H}e_{uvH, vH}\cdot (1\rtimes u)\in\Gamma_{|G/H|}(\tilde{D}\rtimes G)\supset M_{|G/H|}(D\rtimes G) \]
    It is easily seen that $\rho=V\cdot\lambda\cdot V^{-1}$. This proves the equality \eqref{eq:nuHnuuH} and concludes the proof of the proposition.
\end{proof}

\begin{prop}
Let $G$ be a group. Then every morphism $G/H\to G/K$ in $\Or{G}$ induces a (graded) natural transformation $\overline{-\rtimes G/H}\to \overline{-\rtimes G/K}$ of functors $kk^G\tokk$. Moreover, these assemble into a functor $-\rtimes -:kk^G\times \Or{G}\to kk$.
\end{prop}
\begin{proof}
    Let $f:G/H\to G/K$ be a morphism in $\Or{G}$. Clearly, $f$ induces a natural transformation $j\circ\mathcal{R}(-\rtimes G/H)\to j\circ\mathcal{R}(-\rtimes G/K)$ of functors ${G\mathrm{Alg}_\ell}\tokk$. We will prove that this natural transformation is compatible with the excisive homology theory structures. More precisely, let $\mathcal{E}:A\to B\to C$ be an extension in ${G\mathrm{Alg}_\ell}$. We will prove that the following square in $kk$ commutes:
    \begin{equation}\label{eq:deltafunc}\begin{gathered}
    \xymatrix@C=4em{\Omega(\mathcal{R}(C\rtimes G/H))\ar[d]_-{f_*}\ar[r]^-{\partial_{\mathcal{E}\rtimes G/H}} & \mathcal{R}(A\rtimes G/H)\ar[d]^-{f_*} \\
    \Omega(\mathcal{R}(C\rtimes G/K))\ar[r]^-{\partial_{\mathcal{E}\rtimes G/K}} & \mathcal{R}(A\rtimes G/K)}
    \end{gathered}
    \end{equation}
    Supose for a moment that this square commutes. Put $\mathscr{A}=(kk)^I$ where $I$ is the interval category. Then $f$ induces a functor $\nu_f:{G\mathrm{Alg}_\ell}\to\mathscr{A}$ defined by:
    \[D\mapsto \left(f_*:\mathcal{R}(D\rtimes G/H)\to \mathcal{R}(D\rtimes  G/K) \right)\]
    The commutativity of \eqref{eq:deltafunc} implies that $\nu_f$ is a homotopy invariant and $G$-stable $\delta$-functor \cite{loopsthtpy}*{Definition 10.6}. Thus, it factors uniquely through $kk^G$ by universal property of $kk^G$; cf. \cite{loopsthtpy}*{Theorem 10.15}:
    \[\xymatrix{{G\mathrm{Alg}_\ell}\ar[r]^-{j^G}\ar@/_/[dr]_-{\nu_f} & kk^G\ar@{-->}[d]^-{\exists !\,\bar{\nu}_f} \\
    & \mathscr{A}}\]
    The functor $\bar{\nu}_f$ corresponds to the desired natural transformation.
    Let us now show that \eqref{eq:deltafunc} commutes. The morphism $f:G/H\to G/K$ is determined by $f(H)=uK$ for some $u\in G$ with $H\subseteq uKu^{-1}$. We have the following commutative square of $\ell$-linear categories:
    \[\xymatrix@C=4em{A\rtimes G/H\ar[r]^-{f_*} & A\rtimes G/K \\
    A\rtimes H\ar[u]\ar[r]^-{\mathrm{incl}} & A\rtimes uKu^{-1}\ar[u]}\]
    The bottom arrow is an inclusion of algebras. The left and right arrows are the inclusions of the full subcategories whose only objects are $H$ and $uK$, respectively. Upon applying $j\circ\mathcal{R}:{\mathrm{Cat}_\ell}\tokk$ the vertical arrows become isomorphisms and we get:
    \[f_*=\nu_{uK}\circ j(\mathrm{incl})\circ (\nu_H)^{-1}:\mathcal{R}(A\rtimes G/H)\longrightarrow \mathcal{R}(A\rtimes G/K)\]
    Thus, the commutativity of \eqref{eq:deltafunc} is equivalent to that of the outer square in the following diagram in $kk$:
    \[\xymatrix@C=4em{
    \Omega(\mathcal{R}(C\rtimes G/H))\ar[d]_-{\Omega(\nu_H)^{-1}}^-{\cong}\ar[r]^-{\partial_{\mathcal{E}\rtimes G/H}} & \mathcal{R}(A\rtimes G/H)\ar[d]_-{\cong}^-{(\nu_H)^{-1}} \\
    \Omega(C\rtimes H)\ar[r]^-{\partial_{\mathcal{E}\rtimes H}}\ar[d]_-{\Omega j(\mathrm{incl})} & A\rtimes H\ar[d]^-{j(\mathrm{incl})} \\
    \Omega(C\rtimes uKu^{-1})\ar[d]_-{\Omega(\nu_{uK})}^-{\cong}\ar[r]^-{\partial_{\mathcal{E}\rtimes uKu^{-1}}} & A\rtimes uKu^{-1}\ar[d]^-{\nu_{uK}}_-{\cong} \\
    \Omega(\mathcal{R}(C\rtimes G/K))\ar[r]^-{\partial_{\mathcal{E}\rtimes G/K}} & \mathcal{R}(A\rtimes G/K)
    }\]
    Here, the bottom and top squares commute by Proposition \ref{prop:existencertimesGH}. The middle square commutes because it fits into the morphism of triangles induced by the inclusion $H\subseteq uKu^{-1}$.
\end{proof}

\section{A natural adjunction}\label{sec:natadj}
Let $G$ be a group and let $H$ be a finite subgroup of $G$. Recall from Lemma \ref{lem:natisocompoGH} that there is a natural isomorphism $\nu_{H}:-\rtimes H\to \overline{-\rtimes G/H}$ of functors $kk^G\tokk$. If $\frac{1}{|H|}\in\ell$, we have isomorphisms
\begin{equation}\label{eq:natadjdefi}kk^G(A^{(G/H)}, B)\cong kk(A,B\rtimes H)\cong kk(A, \mathcal{R}(B\rtimes G/H)),\end{equation}
where the isomorphism on the right is induced by $\nu_H$ and the one on the left is that of Proposition \ref{prop:adj}. In other words, there is an adjunction:
\begin{equation}\label{eq:natadj}\begin{gathered}\begin{tikzcd}
(-)^{(G/H)}:kk \arrow[r,shift left]
    &
kk^G:\overline{-\rtimes G/H} \arrow[l,shift left]
\end{tikzcd}\end{gathered}\end{equation}
We will show that this adjunction is natural in $G/H$. For the rest of this paper, we assume that $G$ satisfies the following property: \begin{equation}\label{eq:hipG}\tfrac{1}{|H|}\in\ell\text{ for every }H\in\mathcal{F}in\end{equation}

We will prove that for every morphism $f:G/H\to G/K\in\OrFin{G}$ the following square commutes:
\begin{equation}\label{eq:natadjsq}\begin{gathered}\xymatrix{kk(A,\mathcal{R}(B\rtimes G/H))\ar[r]^-{\cong}\ar[d]_-{f_*} & kk^G(A^{(G/H)}, B)\ar[d]^-{f_*} \\
kk(A, \mathcal{R}(B\rtimes G/K))\ar[r]^-{\cong} & kk^G(A^{(G/K)}, B)}\end{gathered}\end{equation}
The commutativity of this square is not obvious a priori since the middle term of \eqref{eq:natadjdefi} is not a functor on $\OrFin{G}$. Let $\alpha\inkk(A, \mathcal{R}(B\rtimes G/H))$ and $f:G/H\to G/K$ in $\OrFin{G}$. The commutativity of \eqref{eq:natadjsq} is equivalent to that of the outer square in the following diagram in $kk$, where $\epsilon_{G/H}$ is the counit of \eqref{eq:natadj}:
\begin{equation}\label{eq:natadjsqexpl}
    \begin{gathered}\xymatrix@C=4em{
    A^{(G/H)}\ar[r]^-{\alpha^{(G/H)}} & \left[\mathcal{R}(B\rtimes G/H)\right]^{(G/H)}\ar[r]^-{\epsilon_{G/H}} & B \\
    A^{(G/K)}\ar[u]^-{A^{(f)}}\ar[r]_-{\alpha^{(G/K)}} & \left[\mathcal{R}(B\rtimes G/H)\right]^{(G/K)}\ar[u]_-{f^*}\ar[r]_-{f_*} & \left[\mathcal{R}(B\rtimes G/K)\right]^{(G/K)}\ar[u]_-{\epsilon_{G/K}}}
    \end{gathered}
\end{equation}
The square on the left clearly commutes. The commutativity of the square on the right follows easily from the following result. 

\begin{lem}\label{lem:natcounit}
Let $G$ be a group satisfying \eqref{eq:hipG}. Then:
\begin{enumerate}
    \item\label{item:defizetaGH} For every $G/H\in\OrFin{G}$ there is a $G$-functor
    \begin{equation}\label{eq:defiZeta}
    \zeta_{G/H}:(B\rtimes G/H)^{(G/H)}\to M_G\otimes B\end{equation}
    that sends $(b\rtimes g)\chi_{sH}\in \hom(uH,vH)=B\otimes\ell[vHu^{-1}]^{(G/H)}$ to:
    \[\sum_{p\in sHv^{-1}}e_{p,pg}\otimes p(b)\]
    \item\label{item:zetacompat} For any $f:G/H\to G/K$ in $\OrFin{G}$, the following square in ${G\mathrm{Cat}_\ell}$ commutes:
    \begin{equation}\label{eq:zetacompat}\begin{gathered}\xymatrix{(B\rtimes G/H)^{(G/H)}\ar[r]^-{\zeta_{G/H}} & M_G\otimes B\\
    (B\rtimes G/H)^{(G/K)}\ar[u]_-{f^*}\ar[r]^-{f_*} & (B\rtimes G/K)^{(G/K)}\ar[u]_-{\zeta_{G/K}}}\end{gathered}\end{equation}
    \item\label{item:couniadj} The counit $\epsilon_{G/H}$ of the adjunction \eqref{eq:natadj} fits into the following commutative diagram in $kk^G$:
    \[\xymatrix@C=5em{
    \left[\mathcal{R}\left(B\rtimes G/H\right)\right]^{(G/H)}\ar@/_1pc/[ddr]_-{\epsilon_{G/H}} & \mathcal{R}\left[\left(B\rtimes G/H\right)^{(G/H)}\right]\ar[d]^-{\mathcal{R}(\zeta_{G/H})}\ar[l]_-{\text{Cor. \ref{coro:rtensor}}}^-{\cong} \\
    & M_G\otimes B\ar[d]^-{\text{zig-zag}}_-{\cong} \\
    & B}
    \]
\end{enumerate}
\end{lem}
\begin{proof}
Let us verify that $\zeta_{G/H}$ as defined in \eqref{item:defizetaGH} is compatible with composition. Let $f_1=(b_1\rtimes g_1)\chi_{sH}\in \mathrm{Hom}(uH, vH)$ and $f_2=(b_2\rtimes g_2)\chi_{tH}\in \mathrm{Hom}(vH, wH)$. We have
\[f_2\circ f_1=\delta_{sH,tH} (b_2g_2(b_1)\rtimes g_2g_1)\chi_{tH}\]
where $\delta_{sH,tH}$ is Kronecker's delta. Then:
\[\zeta_{G/H}(f_2\circ f_1)=\delta_{sH,tH}\sum_{p\in tHw^{-1}}e_{p,pg_2g_1}\otimes p(b_2g_2(b_1))\]
On the other hand, we have:
\begin{align*}\zeta_{G/H}(f_2)\zeta_{G/H}(f_1)&=\left(\sum_{q\in tHw^{-1}}e_{q, qg_2}\otimes q(b_2)\right)\left(\sum_{p\in sHv^{-1}}e_{p, pg_1}\otimes p(b_1)\right)\\
&=\sum_{\substack{p\in sHv^{-1}\\ q\in tHw^{-1}}}e_{q,qg_2}e_{p,pg_1}\otimes q(b_2)p(b_1)\\
&=\delta_{sH,tH}\sum_{q\in tHw^{-1}}e_{q,qg_2g_1}\otimes q(b_2)(qg_2)(b_1)\end{align*}
The appearance of Kronecker's delta in the last line is explained as follows: $qg_2\in tHv^{-1}$ and $p\in sHv^{-1}$ can be equal if and only if $sH=tH$. This shows that $\zeta_{G/H}$ is indeed a well-defined functor.

Let us prove \eqref{item:zetacompat}. The square commutes on objects since $M_G\otimes B$ has only one object. A morphism $f:G/H\to G/K$ is determined by $f(H)=xK$ with $x$ such that $H\subseteq xKx^{-1}$. Let $(b\rtimes g)\chi_{sK}\in \mathrm{Hom}(uH,vH)$. We have:
\begin{align*}
    \zeta_{G/K}\left(f_*\left((b\rtimes g)\chi_{sK}\right)\right)&=\zeta_{G/K}\left((b\rtimes g)\chi_{sK}\right)\\
    &=\sum_{p\in sK(vx)^{-1}}e_{p,pg}\otimes p(b)
\end{align*}
On the other hand, we have:
\begin{align*}
    \zeta_{G/H}\left(f^*\left((b\rtimes g)\chi_{sK}\right)\right)&=\zeta_{G/H}\left(\sum_{tH\in f^{-1}(sK)}(b\rtimes g)\chi_{tH}\right)\\
    &=\sum_{tH\in f^{-1}(sK)}\sum_{p\in tHv^{-1}}e_{p,pg}\otimes p(b)\\
    &=\sum_{p\in sKx^{-1}v^{-1}}e_{p,pg}\otimes p(b)
\end{align*}
Here, the last equality follows from the fact that $sKx^{-1}$ is the disjoint union of $f^{-1}(sK)$. This finishes the proof of \eqref{item:zetacompat}.

Let us prove \eqref{item:couniadj}. It follows from \eqref{eq:natadjdefi} that the counit $\epsilon_{G/H}$ of the adjunction \eqref{eq:natadj} equals $\varepsilon_H\circ \left[(\nu_H)^{-1}\right]^{(G/H)}$, where $\varepsilon_H$ is the counit of \eqref{eq:adjnnat}. Recall the definition of $\varepsilon_H$ from \eqref{eq:couniH}. Then $\epsilon_{G/H}$ equals the following composite in $kk$:
\[\xymatrix@C=4.3em{\left[\mathcal{R}(B\rtimes G/H)\right]^{(G/H)}\ar[r]^-{\left[(\nu_H)^{-1}\right]^{(G/H)}} & (B\rtimes H)^{(G/H)}\ar[r]^-{\psi}_-{\text{\eqref{eq:defipsi}}} & M_G\otimes B\ar[r]^-{\text{zig-zag}}_-{\cong} & B}\]
The statement in \eqref{item:couniadj} now follows from the commutativity of the following diagram, which is straightforward --- indeed, the diagram commutes in ${\mathrm{Alg}_\ell}$:
\[\xymatrix@C=5em{\left[\mathcal{R}(B\rtimes G/H)\right]^{(G/H)} & \mathcal{R}\left[(B\rtimes G/H)^{(G/H)}\right]\ar[l]_-{\text{Cor. \ref{coro:rtensor}}}\ar[d]^-{\mathcal{R}(\zeta_{G/H})} \\
(B\rtimes H)^{(G/H)}\ar[r]^-{\psi}\ar[u]^-{(\nu_H)^{(G/H)}}\ar[ur] & M_G\otimes B}\]
Here the diagonal morphism is the one induced by the inclusion
\[(B\rtimes H)^{(G/H)}\subseteq (B\rtimes G/H)^{(G/H)}\]
as the full subcategory on the object $H$. This finishes the proof.
\end{proof}

We collect the main results of this section in the following theorem.

\begin{thm}\label{thm:adj}
Let $G$ be a group satisfying \eqref{eq:hipG} and let $H$ be a finite subgroup of $G$. We have an adjunction
\[\begin{tikzcd}
(-)^{(G/H)}:kk \arrow[r,shift left]
    &
kk^G:\overline{-\rtimes G/H} \arrow[l,shift left]
\end{tikzcd}\]
\[kk^G(A^{(G/H)}, B)\cong kk(A, \mathcal{R}(B\rtimes G/H))\]
that is natural in $G/H\in\OrFin{G}$. Moreover, the counit of this adjunction is described by Lemma \ref{lem:natcounit}.
\end{thm}


\section{Lifting the adjunction to spectra}\label{sec:bomba}

\subsection{The primitive zig-zag}\label{sec:zzagp} 

Let $A\in{\mathrm{Alg}_\ell}$, $B\in{G\mathrm{Alg}_\ell}$ and $G/N\in\OrFin{G}$. By Theorem \ref{thm:adj} we have an adjunction isomorphism
\begin{equation}\label{eq:natAdjGroups}kk^G(A^{(G/N)}, B)\cong kk(A, B\rtimes G/N)\end{equation}
that is natural in $G/N$.
Let $\mathbb{K}$ and $\mathbb{K}^G$ be, respectively, the spectra representing $kk$-theory and $kk^G$-theory. These were defined in \cite{garku} and in \cite{tesisema}; see Section \ref{sec:sptkkg} for details. We would like to lift the isomorphism \eqref{eq:natAdjGroups} to a natural weak equivalence of spectra:
\[\xymatrix{ \mathbb{K}(A, \mathcal{R}(B\rtimes G/N))\ar@{-->}[r] & \mathbb{K}^G(A^{(G/N)}, B)}
\]
Here, we want the dashed arrow to represent a zig-zag of $\OrFin{G}$-spectra inducing the isomorphism \eqref{eq:natAdjGroups} upon taking homotopy groups. As a starting point, let us recall how to obtain this adjunction. By Lemma \ref{lem:natcounit} \eqref{item:couniadj}, the isomorphism \eqref{eq:natAdjGroups} equals the following composite: 
\begin{equation}\label{eq:decompAdj}\begin{gathered}\xymatrix@C=3.5em{
kk(A, \mathcal{R}(B\rtimes G/N))\ar[r]^-{(-)^{(G/N)}} & kk^G(A^{(G/N)}, \left[\mathcal{R}(B\rtimes G/N)\right]^{(G/N)})\\
 kk^G(A^{(G/N)}, M_{G}B)\ar[d]_-{\rotatebox{90}{$\scriptstyle\cong$}} & kk^G(A^{(G/N)}, \mathcal{R}\left[(B\rtimes G/N)^{(G/N)}\right])\ar[u]^-{\text{Cor. \ref{coro:rtensor}}}_-{\rotatebox{-90}{$\scriptstyle\cong$}}\ar[l]_-{\mathcal{R}(\zeta_{G/N})}^-{\eqref{eq:defiZeta}} \\
kk^G(A^{(G/N)}, M_{G_+}B) & kk^G(A^{(G/N)}, B)\ar[l]_-{\cong}\\
}\end{gathered}\end{equation}
The last two morphisms are easily lifted to spectra. Indeed, the $G$-stability zig-zag \eqref{eq:zigzag} induces a zig-zag of weak equivalences that is clearly natural in $G/N$:
\[\xymatrix{
\mathbb{K}^G(A^{(G/N)}, M_{G}B)\ar[r]^-{\sim} & \mathbb{K}^G(A^{(G/N)}, M_{G_+}B) & \mathbb{K}^G(A^{(G/N)}, B)\ar[l]_-{\sim}
}\]
Lifting the rest of \eqref{eq:decompAdj} is somewhat more delicate. If we simply replace groups by spectra, we get:
\begin{equation}\label{eq:zzagRecortado}
\begin{gathered}\xymatrix@C=3.5em{
\mathbb{K}(A, \mathcal{R}(B\rtimes G/N))\ar[r]^-{(-)^{(G/N)}} & \mathbb{K}^G(A^{(G/N)}, \left[\mathcal{R}(B\rtimes G/N)\right]^{(G/N)})\\
 \mathbb{K}^G(A^{(G/N)}, M_{G}B) & \mathbb{K}^G(A^{(G/N)}, \mathcal{R}\left[(B\rtimes G/N)^{(G/N)}\right])\ar[u]^-{\text{Cor. \ref{coro:rtensor}}}_-{\rotatebox{90}{$\scriptstyle\sim$}}\ar[l]_-{\mathcal{R}(\zeta_{G/N})}^-{\eqref{eq:defiZeta}}}\end{gathered}\end{equation}
This is what we call the \emph{primitive zig-zag}. While the spectra on the left are covariant functors on $\OrFin{G}$, this is not the case for those on the right.
We should start by replacing the latter by covariant functors on $\OrFin{G}$ if we expect a zig-zag that is natural in $G/N$. In the following sections---taking the primitive zig-zag as our model---we proceed to construct a zig-zag of spectra that depends covariantly on $G/N$ and that induces the isomorphism \eqref{eq:natAdjGroups} upon taking homotopy groups.

\subsection{Notation and preliminary definitions}\label{sec:notpreli}
To ease notation, the category $\OrFin{G}$ will be denoted by $\mathcal{O}$ for the rest of this section. Its objects---the orbit spaces corresponding to finite subgroups of $G$---will be denoted by letters $r$, $s$ and $t$.

Let $\mathcal{C}$ be a category and $\Gamma$ be a small category. We will write $B(\Gamma, \mathcal{C})$ for the category $\mathcal{C}^{\Gamma^\mathit{op}\times\Gamma}$ of bifunctors $\Gamma^\mathit{op}\times\Gamma\to\mathcal{C}$. Let $f:\Gamma\to \Lambda$ be a functor between small categories. Then we can restrict along $f$ either of the variables of a bifunctor $\Lambda^\mathit{op}\times\Lambda\to\mathcal{C}$, or both of them, as shown by the following commutative diagram of categories:
\[\xymatrix{
B(\Lambda,\mathcal{C})\ar[r]^-{f^*}\ar[d]_-{f^*}\ar[dr]^-{f^\star} & \mathcal{C}^{\Lambda^\mathit{op}\times\Gamma}\ar[d]^-{f^*} \\
\mathcal{C}^{\Gamma^\mathit{op}\times\Lambda}\ar[r]^-{f^*} &
B(\Gamma,\mathcal{C})
}\]
Here, $f^*$ denotes restriction of one of the variables (either the covariant or the contravariant one) and $f^\star$ denotes restriction of both variables.

Define functors $J\in\mathrm{Sp}^\mathcal{O}$ and $M, L\in B(\mathcal{O},\mathrm{Sp})^\mathcal{O}$ by
\begin{align}
    J(t)&:=\mathbb{K}(A, \mathcal{R}(B\rtimes t)) \nonumber\\
    L_t(s,r)&:=\mathbb{K}^G(A^{(t)}, \mathcal{R}[(B\rtimes r)^{(s)}]) \label{eq:defisFuntores}\\
    M_t(s,r)&:=\mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r)]^{(s)}) \nonumber
\end{align}
for $r,s,t\in\mathcal{O}$. The $kk^G$-equivalence $\mathcal{R}[(B\rtimes r)^{(s)}]\to \left[\mathcal{R}(B\rtimes r)\right]^{(s)}$ of Corollary \ref{coro:rtensor} induces, upon applying $\mathbb{K}^G(A^{(t)}, -)$, a natural transformation $\psi:L\to M$ that is an objecwise weak equivalence of spectra. With this notation, the primitive zig-zag \eqref{eq:zzagRecortado} becomes:
\begin{equation}\label{eq:zzagPrimit}\xymatrix{
J(t) \ar[r]^-{(-)^{(t)}} &
M_t(t, t) & L_t(t,t)\ar[l]^-{\sim}_-{\psi}\ar[r]^-{\mathcal{R}(\zeta_t)} &
\mathbb{K}^G(A^{(t)}, M_G B)
}\end{equation}

\subsection{Coends enter the game}\label{sec:coend}
Fix $t\in\mathcal{O}$. The commutativity of \eqref{eq:zetacompat} suggests that the morphism induced by $\mathcal{R}(\zeta_t)$ in \eqref{eq:zzagPrimit} could be replaced by a morphism from a certain coend, as we proceed to explain. If $f:r\to s$ is a morphism in $\mathcal{O}$, the following square commutes by Lemma \ref{lem:natcounit} \eqref{item:zetacompat}:
\[
\xymatrix{(B\rtimes r)^{(r)}\ar[r]^-{\zeta_r} & M_G\otimes B\\
    (B\rtimes r)^{(s)}\ar[u]_-{f^*}\ar[r]^-{f_*} & (B\rtimes s)^{(s)}\ar[u]_-{\zeta_{s}}}
\]
Upon applying $\mathbb{K}^G(A^{(t)}, \mathcal{R}(-))$, we get a commutative diagram:
\begin{equation}\label{eq:sqZetaK1}\begin{gathered}\xymatrix{
L_t(r,r)\ar[r]^-{\zeta_r} & \mathbb{K}^G(A^{(t)}, M_GB)\\
L_t(s,r)\ar[u]^-{f^*}\ar[r]^-{f_*} & L_t(s,s)\ar[u]_-{\zeta_s}
}\end{gathered}\end{equation}
For reasons that will become clear later on (see Remark \ref{rem:choiceOfSlice1} and Lemma \ref{lem:nat:isofun}) we will take coends over the slice category $\slice{\mathcal{O}}{t}$ of orbit spaces over $t$. We will denote by $u_t$ the forgetful functor $\slice{\mathcal{O}}{t}\to \mathcal{O}$. Let now $f:\alpha\to\beta$ be a morphism in $\slice{\mathcal{O}}{t}$, where $\alpha:r\to t$ and $\beta:s\to t$. Then \eqref{eq:sqZetaK1} equals the square:
\[\xymatrix{
\left[\Ustar{t} L_t \right](\alpha, \alpha)\ar[r]^-{\zeta_r} & \mathbb{K}^G(A^{(t)}, M_GB)\\
\left[\Ustar{t} L_t \right](\beta, \alpha)\ar[u]^-{f^\star }\ar[r]^-{f_*} &
\left[\Ustar{t} L_t \right](\beta, \beta)\ar[u]_-{\zeta_s}
}\]
By the universal property of the coend, there is a unique morphism $\zeta$ making the following triangle commute for all objects $\alpha:r\to t$ of $\slice{\mathcal{O}}{t}$:
\begin{equation}\label{eq:defiZetaNat}\begin{gathered}\xymatrix{
\intg{t}\Ustar{t} L_t \ar[r]^-{\zeta} & \mathbb{K}^G(A^{(t)}, M_GB) \\
\left[\Ustar{t} L_t \right](\alpha, \alpha)\ar[u]^-{\mathit{can}_\alpha}\ar@/_1.5pc/[ur]_-{\zeta_r}
}\end{gathered}\end{equation}
Here the vertical morphism is the structural morphism into the coend corresponding to $\alpha$.
In the next section we will prove that $\zeta$ depends covariantly on $t\in\mathcal{O}$.



\subsection{Defining \texorpdfstring{$\mathcal{O}$}{O}-spectra as objectwise coends} \label{sec:coend2} Let us show that the morphisms $\zeta$ of \eqref{eq:defiZetaNat} assemble, for varying $t$, into a morphism of $\mathcal{O}$-spectra. We first prove some preliminary lemmas.

\begin{lem}\label{lem:natcoend}
Let $\mathcal{C}$ be a cocomplete category, $f:\Gamma\to\Lambda$ be a functor between small categories and $T\inB(\Lambda ,\mathcal{C})$. Then there is a unique morphism
\begin{equation}\label{eq:natcoend}\xymatrix{\int^{\Gamma} f^\star T\ar[r] & \int^{\Lambda}  T}\end{equation}
making the following square in $\mathcal{C}$ commute, for every object $\gamma$ of $\Gamma$:
\[\xymatrix{
\int^{\Gamma} f^\star T\ar[r] & \int^{\Lambda}  T \\
(f^\star T)(\gamma, \gamma)\ar@{=}[r]\ar[u]^-{\mathit{can}_{\gamma}} & T(f(\gamma), f(\gamma))\ar[u]_-{\mathit{can}_{f(\gamma)}}\ar[u]
}\]
Here the vertical arrows are the structural morphisms into the coends. Moreover, \eqref{eq:natcoend} is natural in $T$.
\end{lem}
\begin{proof}
This is immediate from the universal property of coends.
\end{proof}

\begin{rem}\label{rem:natcoend2}
For composable functors $\Gamma\xrightarrow{f}\Lambda\xrightarrow{g}\Sigma$ and $T\inB(\Sigma,\mathcal{C})$, the morphisms \eqref{eq:natcoend} clearly fit into the following commutative triangle:
\[\xymatrix{
\int^\Gamma (g\circ f)^\star T=\int^\Gamma f^\star g^\star T\ar[r] & \int^\Lambda g^\star T\ar[d] \\
& \int^\Sigma T
\ar@/_1pc/ "1,1"+<-2em,-0.8em>;"2,2"
}\]
\end{rem}

\begin{lem}\label{lem:nat}
Let $\mathcal{C}$ be a cocomplete category and $\Gamma$ be a small category. Then there is a functor $\funcoend:B(\Gamma,\mathcal{C})^{\Gamma}\to \mathcal{C}^\Gamma$ described as follows.
\begin{enumerate}
    \item\label{item:lemnat1} Let $V$ be an object of $B(\Gamma,\mathcal{C})^{\Gamma}$, $t\in\Gamma\mapsto V_t\in B(\Gamma,\mathcal{C})$. For $t\in\Gamma$, we have:
    \[\funcoend(V)(t)=\textstyle\int^{\slice{\Gamma}{t}} \Ustar{t}V_t\]
    For a morphism $f:t\to t'$ in $\Gamma$, $\funcoend(V)(f)$ equals the composite:
    \[\xymatrix@C=1.7em{
    \int^{\slice{\Gamma}{t}} \Ustar{t}V_t \ar[r]^-{V_f} &
    \int^{\slice{\Gamma}{t}} \Ustar{t}V_{t'}=\int^{\slice{\Gamma}{t}} f^\star\Ustar{t'}V_{t'} \ar[r]^-{\eqref{eq:natcoend}} & \int^{\slice{\Gamma}{t'}}\Ustar{t'}V_{t'}}\]
    \item For a morphism $h:V\to W$ in $B(\Gamma,\mathcal{C})^\Gamma$, the natural transformation $\funcoend(h)$ has components:
    \[\xymatrix@C=6em{\int^{\slice{\Gamma}{t}}\Ustar{t}V_t\ar[r]^-{\int^{\slice{\Gamma}{t}}\Ustar{t}h_t} & \int^{\slice{\Gamma}{t}}\Ustar{t}W_t}\]
\end{enumerate}
\end{lem}
\begin{proof}
The fact that the equalities in \eqref{item:lemnat1} indeed define a functor $\funcoend(V)\in\mathcal{C}^\Gamma$ boils down to the naturality of \eqref{eq:natcoend} and Remark \ref{rem:natcoend2}. The fact that $\funcoend(h)$ is indeed a natural transformation follows as well from the naturality of \eqref{eq:natcoend}.
\end{proof}

\begin{lem}\label{lem:natZeta}
The morphism $\zeta$ defined by \eqref{eq:defiZetaNat} is a morphism of $\mathcal{O}$-spectra.
\end{lem}
\begin{proof}
The codomain of $\zeta$ is clearly an $\mathcal{O}$-spectrum. Its domain is an $\mathcal{O}$-spectrum as well; indeed, it is $\funcoend(t\mapsto L_t)$
where $\mathscr{C}$ is the functor of Lemma \ref{lem:nat}. Let $f:t\to t'$ be a morphism in $\mathcal{O}$. We claim that following square commutes:
\[\xymatrix{
\intg{t} \Ustar{t} L_t\ar[r]^-{\zeta}\ar[d]_-{f} & \mathbb{K}^G(A^{(t)}, M_GB)\ar[d]^-{f} \\
\intg{t'}\Ustar{t'} L_{t'}\ar[r]^-{\zeta} & \mathbb{K}^G(A^{(t')}, M_GB)
}\]
Indeed, by the universal property of the coend, it suffices to show that the square commutes when precomposed with the structural morphisms
\[\left[\Ustar{t} L_t\right](\alpha, \alpha)\to \textstyle\intg{t}\Ustar{t} L_t\]
for every object $\alpha:r\to t$ of $\slice{\mathcal{O}}{t}$. Upon precomposing with the latter we get the following square, that clearly commutes:
\[\xymatrix{
\mathbb{K}^G(A^{(t)}, \mathcal{R}\left[(B\rtimes r)^{(r)}\right])\ar[r]^-{\zeta_{r}}\ar[d]_-{f} & \mathbb{K}^G(A^{(t)}, M_GB)\ar[d]^-{f} \\
\mathbb{K}^G(A^{(t')}, \mathcal{R}\left[(B\rtimes r)^{(r)}\right])\ar[r]^-{\zeta_{r}} & \mathbb{K}^G(A^{(t')}, M_GB)
}\]
This proves the lemma.
\end{proof}

\subsection{The morphism \texorpdfstring{$\varphi$}{phi}}\label{sec:morfi}
In this section we define a morphism $\varphi$ that will be part of the natural zig-zag of Theorem \ref{thm:ZZAG}.
Let $\cte:\mathrm{Sp}^\mathcal{O}\toB(\mathcal{O}, \mathrm{Sp})$ be the functor that adds a constant contravariant variable, defined by
\begin{equation}\label{eq:defiDelta}\cte[F](s,r)=F(r)\end{equation}
for $F\in\mathrm{Sp}^\mathcal{O}$ and $r,s\in\mathcal{O}$. Let $\Uast{t}:B(\mathcal{O}, \mathrm{Sp})\to \mathrm{Sp}^{(\slice{\mathcal{O}}{t})^\mathit{op} \times \mathcal{O}}$ be the restriction of the contravariant variable along the forgetful functor. Recall the definitions of $J\in\mathrm{Sp}^\mathcal{O}$ and $M_t\inB(\mathcal{O}, \mathrm{Sp})$ from \eqref{eq:defisFuntores}. Define a morphism
\begin{equation}\label{eq:phiSharp}\varphi^\sharp:\Uast{t}\cte[J]\to \Uast{t}M_t\end{equation}
as follows. For objects $r$ of $\mathcal{O}$ and $\alpha:s\to t$ of $\slice{\mathcal{O}}{t}$, let
\begin{equation}\label{eq:defiPhi}\varphi^\sharp_{(\alpha, r)}:\left[\Uast{t}\cte[J]\right](\alpha, r)\to \left[\Uast{t}M_t\right](\alpha, r)\end{equation}
be the following composition:
\[
\xymatrix{\mathbb{K}(A, \mathcal{R}(B\rtimes r)) \ar[r]^-{(-)^{(t)}} &
\mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r)]^{(t)})\ar[r]^-{\alpha^*}
& \mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r)]^{(s)})}
\]
It is easily verified that $\varphi^\sharp$ is a natural transformation of bifunctors. Indeed, for morphisms $f:r\to r'$ in $\mathcal{O}$ and $g:\alpha\to\alpha'$ in $\slice{\mathcal{O}}{t}$, the following diagrams commute:

\[
\xymatrix@C=-1em{&\mathbb{K}(A, \mathcal{R}(B\rtimes r))\ar@/_2.5pc/[ddl]_-{\varphi^\sharp_{(\alpha, r)}} \ar[d]^{(-)^{(t)}} \ar@/^2.5pc/[ddr]^-{\varphi^\sharp_{(\alpha', r)}}& \\
&\mathbb{K}^{G}(A^{(t)},[\mathcal{R}(B\rtimes r)]^{(t)})\ar[dr]^{(\alpha')^{*}} \ar[dl]_{\alpha^{*}}&\\
\mathbb{K}^{G}(A^{(t)},[\mathcal{R}(B\rtimes r)]^{(s)}) 
&& \mathbb{K}^{G}(A^{(t)},[\mathcal{R}(B\rtimes r)]^{(s')})\ar[ll]^-{g^{*}}
}
\]

\[
\xymatrix@C=3em{
\mathbb{K}(A, \mathcal{R}(B\rtimes r))\ar@/_6.7pc/[dd]_(.25){\varphi^\sharp_{(\alpha, r)}}\ar[d]^-{(-)^{(t)}}\ar[r]^-{f_*} & \mathbb{K}(A, \mathcal{R}(B\rtimes r'))\ar[d]_-{(-)^{(t)}}\ar@/^6.7pc/[dd]^(.25){\varphi^\sharp_{(\alpha, r')}} \\
\mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r)]^{(t)})\ar[d]^-{\alpha^*}\ar[r]^-{f_*} & \mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r')]^{(t)})\ar[d]_-{\alpha^*} \\
\mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r)]^{(s)})\ar[r]^-{f_*} & \mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r')]^{(s)})
}
\]

\begin{rem}\label{rem:choiceOfSlice1}
The definition of the components of $\varphi^\sharp$ \eqref{eq:defiPhi} makes use of the structural morphism $\alpha:s\to t$ and it is not clear how to define $\varphi^\sharp$ as a morphism $\cte[J]\to M_t$ in $B(\mathcal{O}, \mathrm{Sp})$. This is one of the reasons that motivated our choice of $\slice{\mathcal{O}}{t}$ as the indexing category for coends.
\end{rem}

\begin{cons}\label{const:nat2}
By Remark \ref{rem:resExt} we have, for each $t\in\mathcal{O}$, a pair of adjoint functors:
\begin{equation}\label{eq:adjResPF}\xymatrix{
\Uexcl{t}:\mathrm{Sp}^{(\slice{\mathcal{O}}{t})^\mathit{op}\times \mathcal{O}}\ar@<.5ex>[r]\ar@<-.5ex>@{<-}[r] & B(\mathcal{O}, \mathrm{Sp}):\Uast{t}
}\end{equation}
Let $f:t\to t'$ be a morphism in $\mathcal{O}$ and write $f^*:\mathrm{Sp}^{(\slice{\mathcal{O}}{t'})^\mathit{op}\times\mathcal{O}}\to\mathrm{Sp}^{(\slice{\mathcal{O}}{t})^\mathit{op}\times\mathcal{O}}$ for the restriction of the contravariant variable along $f:\slice{\mathcal{O}}{t}\to\slice{\mathcal{O}}{t'}$. Note that $f^*\circ \Uast{t'}=\Uast{t}$ and consider the following diagram of solid arrows for $F\in\mathrm{Sp}^\mathcal{O}$ and $H\inB(\mathcal{O}, \mathrm{Sp})$:
\[\xymatrix{
B(\mathcal{O},\mathrm{Sp})\left(\Uexcl{t'}\Uast{t'}\cte[F], H\right)\ar[r]^-{\cong}\ar@{-->}[d] &
\mathrm{Sp}^{(\slice{\mathcal{O}}{t'})^\mathit{op}\times\mathcal{O}}\left(\Uast{t'}\cte[F], \Uast{t'}H\right)\ar[d]^-{f^*} \\
B(\mathcal{O}, \mathrm{Sp})\left(\Uexcl{t}\Uast{t}\cte[F], H\right)\ar[r]^-{\cong} &
\mathrm{Sp}^{(\slice{\mathcal{O}}{t})^\mathit{op}\times\mathcal{O}}\left(\Uast{t}\cte[F], \Uast{t}H\right)
}\]
Let the dashed arrow complete the diagram to a commutative square. By the Yoneda Lemma, the dashed arrow is induced by precomposition with a unique morphism $\Uexcl{t}\Uast{t}\cte[F]\to \Uexcl{t'}\Uast{t'}\cte[F]$. The latter, for varying $f$, assemble into a functor $\mathcal{O}\toB(\mathcal{O}, \mathrm{Sp})$, $t\mapsto \Uexcl{t}\Uast{t}\cte[F]$. This construction is, moreover, clearly natural in $F$, so that we get a functor:
\[\xymatrix@R=0em{\mathrm{Sp}^\mathcal{O}\ar@{->}[r] & B(\mathcal{O},\mathrm{Sp})^\mathcal{O} \\
F\ar@{|->}[r] & (t\mapsto \Uexcl{t}\Uast{t}\cte[F])}\]
\end{cons}

\begin{rem}\label{rem:constr}
Let $F\in\mathrm{Sp}^\mathcal{O}$. Later on, it will be useful to have an explicit description of the functor $\mathcal{O}\to B(\mathcal{O}, \mathrm{Sp})$, $t\mapsto \Uexcl{t}\Uast{t}\cte[F]$ mentioned in Construction \ref{const:nat2}. Let us first describe the bifunctor $\Uexcl{t}\Uast{t}\cte[F]\inB(\mathcal{O},\mathrm{Sp})$ for fixed $t\in\mathcal{O}$. For $r,s\in\mathcal{O}$ we have:
\[\left[\Uexcl{t}\Uast{t}\cte[F]\right](s,r)=
\coprod_{\alpha\in\mathcal{O}(s,t)}\left[\Uast{t}\cte[F]\right](\alpha, r)=
\coprod_{\mathcal{O}(s,t)} F(r)\]
For a morphism $f:r\to r'$ in $\mathcal{O}$, the induced morphism $f_*=\left[\Uexcl{t}\Uast{t}\cte[F]\right](s,f)$ equals the morphism:
\[\coprod F(f):\coprod_{\mathcal{O}(s,t)} F(r)\to \coprod_{\mathcal{O}(s,t)} F(r')\]
For a morphism $g:s'\to s$ in $\mathcal{O}$, the induced morphism $g^*=\left[\Uexcl{t}\Uast{t}\cte[F]\right](g,r)$ is the unique morphism making the following triangle commute for all $\beta\in\mathcal{O}(s,t)$:
\[\xymatrix@R=1.5em@C=4em{F(r)\ar[r]^-{\mathit{can}_\beta}\ar@/_1pc/[dr]_-{\mathit{can}_{\beta\circ g}} &
\displaystyle\coprod_{\mathcal{O}(s,t)}F(r)\ar[d]^-{g^*} \\
& \displaystyle\coprod_{\mathcal{O}(s',t)}F(r)
}\]
Now let $h:t\to t'$ be a morphism in $\mathcal{O}$. Then the components of the induced natural transformation $h_*:\Uexcl{t}\Uast{t}\cte[F]\to \Uexcl{t'}\Uast{t'}\cte[F]$ are the unique morphisms making the following triangle commute for all $\beta\in\mathcal{O}(s,t)$:
\[\xymatrix@R=1.5em@C=4em{F(r)\ar[r]^-{\mathit{can}_\beta}\ar@/_1pc/[dr]_-{\mathit{can}_{h\circ \beta}} &
\displaystyle\coprod_{\mathcal{O}(s,t)}F(r)\ar[d]^-{h_*} \\
& \displaystyle\coprod_{\mathcal{O}(s,t')}F(r)
}\]
\end{rem}

\begin{lem}
Fix $t\in\mathcal{O}$ and let $\varphi^\sharp:\Uast{t}\cte[J]\to \Uast{t}M_t$ be the morphism in $\mathrm{Sp}^{(\slice{\mathcal{O}}{t})^\mathit{op}\times\mathcal{O}}$ defined in \eqref{eq:phiSharp}. Under the adjunction \eqref{eq:adjResPF}, $\varphi^\sharp$ corresponds to a morphism $\varphi:\Uexcl{t}\Uast{t}\cte[J]\to M_t$ in $B(\mathcal{O},\mathrm{Sp})$. 
Explicitly, for $r,s\in\mathcal{O}$, the component $\varphi_{(s,r)}:[\Uexcl{t}\Uast{t}\delta_J](s,r)\to M_t(s,r)$ is the unique morphism making the following triangle commute for every $\alpha\in\mathcal{O}(s,t)$:
\[\xymatrix{\displaystyle\coprod_{\mathcal{O}(s,t)}J(r)\ar[r]^-{\varphi_{(s,r)}} &
M_t(s,r) \\
J(r)\ar[u]^-{\mathit{can}_{\alpha}}\ar@/_1pc/[ur]_-{\varphi^\sharp_{(\alpha,r)}} &
}\]
Then the latter, for varying $t$, assemble into a morphism in $B(\mathcal{O},\mathrm{Sp})^\mathcal{O}$---where the domain of $\varphi$ is considered as an object of $B(\mathcal{O},\mathrm{Sp})^\mathcal{O}$ as explained in Construction \ref{const:nat2}.
\end{lem}
\begin{proof}
Let $f:t\to t'$ be a morphism in $\mathcal{O}$. We have to show that the following square in $B(\mathcal{O},\mathrm{Sp})$ commutes:
\[\xymatrix{\Uexcl{t}\Uast{t}\cte[J]\ar[r]^-{\varphi}\ar[d]_-{f_*} &
M_t\ar[d]^-{f_*} \\
\Uexcl{t'}\Uast{t'}\cte[J]\ar[r]^-{\varphi} &
M_{t'}
}\]
By Remark \ref{rem:constr}, for $r,s\in\mathcal{O}$, we have $[\Uexcl{t}\Uast{t}\cte[J]](s,r)=\coprod_{\mathcal{O}(s,t)}J(r)$. 
Thus, it suffices to show that the following square commutes for $r,s\in\mathcal{O}$:
\[\xymatrix{\displaystyle\coprod_{\mathcal{O}(s,t)}J(r)\ar[r]^-{\varphi}\ar[d]_-{f_*} &
M_t(s,r)\ar[d]^-{f_*} \\
\displaystyle\coprod_{\mathcal{O}(s,t')}J(r)\ar[r]^-{\varphi} &
M_{t'}(s,r)
}\]
By the universal property of the coproduct it suffices to show that this square commutes when precomposed with every structural morphism into the coproduct in the upper left corner. Let $\alpha\in\mathcal{O}(s,t)$. Upon precomposing the latter square with $\mathit{can}_\alpha:J(r)\to \coprod_{\mathcal{O}(s,t)}J(r)$ we get:
\[\xymatrix{J(r)\ar[r]^-{\varphi^\sharp_{(\alpha,r)}}\ar@/_1pc/[dr]_-{\varphi^\sharp_{(f\circ \alpha,r)}} & 
M_t(s,r)\ar[d]^-{f_*} \\
& M_{t'}(s,r)
}\]
Unravelling the definition of $\varphi^\sharp$, this triangle becomes:
\[\xymatrix@C=1.3em{
\mathbb{K}(A, \mathcal{R}(B\rtimes r))\ar[r]^-{(-)^{(t)}}\ar[d]_-{(-)^{(t')}} &
\mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r)]^{(t)})\ar[d]^-{f_*}\ar[r]^-{\alpha^*} &
\mathbb{K}^G(A^{(t)}, [\mathcal{R}(B\rtimes r)]^{(s)})\ar[d]^-{f_*} \\
\mathbb{K}^G(A^{(t')}, [\mathcal{R}(B\rtimes r)]^{(t')})\ar[r]^-{f^*} &
\mathbb{K}^G(A^{(t')}, [\mathcal{R}(B\rtimes r)]^{(t)})\ar[r]^-{\alpha^*} &
\mathbb{K}^G(A^{(t')}, [\mathcal{R}(B\rtimes r)]^{(s)})
}\]
The square on the right clearly commutes. The one on the left commutes by Lemma \ref{lem:sqSptComm}.
\end{proof}

\subsection{Model structure on categories of bifunctors}\label{sec:modcatbi}
Let $\mathcal{C}$ be a model category and let $\Gamma$ be a small category. In this section, we endow $B(\Gamma, \mathcal{C})$ with a model structure that will allow us to build models for the \emph{homotopy coends} of certain morphisms of bifunctors. We will need these later on to define the morphisms going backwards in the zig-zag of $\mathcal{O}$-spectra of Theorem \ref{thm:ZZAG}.

\begin{prop}[\cite{luriehtt}*{Proposition A.2.8.2}]\label{prop:htt}Let $\mathcal{C}$ be a combinatorial model category and let $\Gamma$ be a small category. Then there exist two combinatorial model structures on $\mathcal{C}^\Gamma$:
\begin{itemize}
    \item The \emph{injective model structure}, denoted $\mathcal{C}^{\Gamma}_\mathrm{inj}$, where weak equivalences and cofibrations are defined objectwise.
    \item The \emph{projective model structure}, denoted $\mathcal{C}^{\Gamma}_\mathrm{proj}$, where weak equivalences and fibrations are defined objectwise.
\end{itemize}
\end{prop}

We will always consider $B(\Gamma,\mathcal{C})$ as a model category with the structure $(\mathcal{C}_\mathrm{inj}^{\Gamma^\mathit{op}})_\mathrm{proj}^{\Gamma}$ whenever this structure exists. The model structure on $B(\Gamma, \mathrm{Sp})$ exists for any $\Gamma$ by Lemma \ref{lem:sptCombi} and Proposition \ref{prop:htt}.

\begin{thm}[\cite{daneses}*{Theorem 4.1}]\label{thm:coendquillen}
Let $\mathcal{C}$ be a model category and $\Gamma$ be a small category such that the model structure on $B(\Gamma, \mathcal{C})$ exists. Then the functor
\begin{equation}\textstyle\label{eq:thmcoend}\int^{\Gamma}:B(\Gamma, \mathcal{C})\to \mathcal{C}\end{equation}
is a left Quillen functor.
\end{thm}
\begin{proof}
This is \cite{daneses}*{Theorem 4.1}; we sketch the proof for completeness. For $c\in\mathcal{C}$, define $R(c):\Gamma^\mathit{op}\times\Gamma\to\mathcal{C}$ by:
\[R(c)(s,r):=\prod_{\alpha\in\Gamma(r, s)}c\]
For morphisms $f:r\to r'$ and $g:s'\to s$ in $\Gamma$, let $f_*$ and $g^*$ be the unique morphisms making the following diagrams commute for all $\alpha\in\Gamma(r',s)$ and all $\beta\in\Gamma(r,s')$:
\[
\xymatrix@C=4em{R(c)(s,r)\ar[r]^-{f_*}\ar@/_0.5pc/[dr]_-{\mathit{can}_{\alpha\circ f}} & R(c)(s,r')\ar[d]^-{\mathit{can}_{\alpha}} \\
& c
}
\hspace{2em}
\xymatrix@C=4em{R(c)(s,r)\ar[r]^-{g^*}\ar@/_0.5pc/[dr]_-{\mathit{can}_{g\circ\beta}} & R(c)(s',r)\ar[d]^-{\mathit{can}_{\beta}} \\
& c
}
\]
It is easily verified that $R:\mathcal{C}\to\mathcal{C}^{\Gamma^\mathit{op}\times\Gamma}$ is right adjoint to $\int^\Gamma:\mathcal{C}^{\Gamma^\mathit{op}\times\Gamma}\to\mathcal{C}$. Thus, proving the theorem is equivalent to showing that
\[R:\mathcal{C}\toB(\Gamma, \mathcal{C})\]
preserves fibrations and trivial fibrations. Let $c\to c'$ be a (trivial) fibration in $\mathcal{C}$. To prove that $R(c)\to R(c')$ is a (trivial) fibration in $B(\Gamma, \mathcal{C})=(\mathcal{C}^{\Gamma^\mathit{op}}_\mathrm{inj})^{\Gamma}_\mathrm{proj}$, it suffices to show that $R(c)(-,r)\to R(c')(-,r)$ is a (trivial) fibration in $\mathcal{C}^{\Gamma^\mathit{op}}_\mathrm{inj}$ for every $r\in\Gamma$. But the latter holds since, for every $r$, there is a Quillen adjunction
\[\mathrm{ev}_r:\mathcal{C}^{\Gamma^\mathit{op}}_\mathrm{inj}\rightleftarrows \mathcal{C}:R(-)(-,r)\]
where $\mathrm{ev}_r$ is the evaluation at $r$ \cite{daneses}*{Corollary 2.3 (iii)}. Indeed, this adjunction is Quillen since $\mathrm{ev}_r$ clearly preserves cofibrations and trivial cofibrations.
\end{proof}

\begin{lem}[\cite{luriehtt}*{Proposition A.2.8.7}]\label{lem:resLeftQuillen1}
Let $\mathcal{C}$ be a model category and let $f:\Gamma\to\Lambda$ be a functor. Then the restriction functor $f^*:\mathcal{C}^\Lambda\to\mathcal{C}^\Gamma$ fits into the following Quillen adjunctions, whenever the model structures in question exist:
\begin{enumerate}
    \item $f^*:\mathcal{C}^\Lambda_\mathrm{inj}\rightleftarrows \mathcal{C}^\Gamma_\mathrm{inj}:f_*$
    \item $f_!:\mathcal{C}^\Gamma_\mathrm{proj}\rightleftarrows \mathcal{C}^\Lambda_\mathrm{proj}:f^*$
\end{enumerate}
\end{lem}

\begin{rem}\label{rem:resExt}
Let $\mathcal{C}$ be a category with small coproducts and let $\Gamma$ be a small category. Fix $t\in\Gamma$ and let $u_t:\slice{\Gamma}{t}\to \Gamma$ be the forgetful functor. Then there is an adjunction:
\[\begin{tikzcd}
\Uexcl{t}:\mathcal{C}^{(\slice{\Gamma}{t})^\mathit{op}} \arrow[r,shift left]
    &
\mathcal{C}^{\Gamma^\mathit{op}}:\Uast{t} \arrow[l,shift left]
\end{tikzcd}\]
Moreover, the pushforward functor $\Uexcl{t}$ can be explicitely described as follows. For $s\in\Gamma$, we have:
\[\left[\Uexcl{t} F\right](s)=\coprod_{\alpha\in\mathrm{Hom}(s, t)}F(\alpha)\]
For a morphism $g:s'\to s$ in $\Gamma$, $\left[\Uexcl{t} F\right](g)$ is the unique morphism making the following squares commute, where the vertical arrows are the structural morphisms into the coproducts:
\[\xymatrix@C=7em{
\displaystyle\coprod_{\alpha\in\mathrm{Hom}(s, t)}F(\alpha)\ar[r]^-{\left[\Uexcl{t} F\right](g)} & \displaystyle\coprod_{\alpha'\in\mathrm{Hom}(s', t)}F(\alpha') \\
F(\beta)\ar[u]^-{\mathit{can}_\beta}\ar[r]^-{F(g)} & F(\beta\circ g)\ar[u]_-{\mathit{can}_{\beta\circ g}}
}\]
\end{rem}

\begin{lem}[cf. \cite{luriehtt}*{Lemma A.2.8.10}]\label{lem:resLeftQuillen2}
Let $\mathcal{C}$ be a model category, $\Gamma$ be a small category, fix $t\in\Gamma$, and let $u_t:\slice{\Gamma}{t}\to\Gamma$ be the forgetful functor from the slice category. Then the following adjunctions are Quillen adjunctions, whenever the model structures in question exist:
\begin{enumerate}
    \item\label{item:adj1} $\Uast{t}:\mathcal{C}^{\Gamma}_\mathrm{proj}\rightleftarrows \mathcal{C}^{\slice{\Gamma}{t}}_\mathrm{proj}:\Uastd{t}$
    \item\label{item:adj2} $\Uexcl{t}:\mathcal{C}^{(\slice{\Gamma}{t})^\mathit{op}}_\mathrm{inj}\rightleftarrows \mathcal{C}^{\Gamma^\mathit{op}}_\mathrm{inj}:\Uast{t}$
\end{enumerate}
\end{lem}
\begin{proof}
To prove \eqref{item:adj2} is a Quillen adjunction, let us show that $\Uexcl{t}$ preserves cofibrations and trivial cofibrations. Recall from Remark \ref{rem:resExt} that, for $F\in\mathcal{C}^{(\slice{\Gamma}{t})^\mathit{op}}$ and $s\in\Gamma$, we have:
\[\left[\Uexcl{t} F\right](s)=\coprod_{\alpha\in\Gamma(s, t)}F(\alpha)\]
Let $\eta:F\to F'$ be a morphism in $\mathcal{C}^{(\slice{\Gamma}{t})^\mathit{op}}$. For $s\in\Gamma$, $\Uexcl{t}(\eta)(s)$ is the coproduct of the morphisms:
\[\{\eta(\alpha):F(\alpha)\to F'(\alpha)\}_{\alpha\in\Gamma(s, t)}\]
If $\eta$ is a cofibration (resp. a trivial cofibration) in $\mathcal{C}^{(\slice{\Gamma}{t})^\mathit{op}}_\mathrm{inj}$, the latter are cofibrations (resp. trivial cofibrations) in $\mathcal{C}$ and, thus, $\Uexcl{t}(\eta)(s)$ is again a cofibration (resp. trivial cofibration) in $\mathcal{C}$. Since this holds for every $s\in\Gamma$, it follows that $\Uexcl{t}(\eta)$ is a cofibration (resp. a trivial cofibration) in $\mathcal{C}^{\Gamma^\mathit{op}}_\mathrm{inj}$.
The proof of \eqref{item:adj1} is dual to that of \eqref{item:adj2}, using the fact that
\[\left[\Uastd{t} F\right](s)=\prod_{\alpha\in\Gamma(t, s)}F(\alpha)\]
for $F\in \mathcal{C}^{\slice{\Gamma}{t}}$ and $s\in\Gamma$.
\end{proof}

\begin{lem}[\cite{luriehtt}*{Remark A.2.8.6}]\label{lem:htt}
Let $F:\mathcal{C}\rightleftarrows \mathcal{D}:U$ be a Quillen adjunction of combinatorial model categories and let $\Gamma$ be a small category. Then composition with $F$ and $U$ determines a Quillen adjunction
\[F^{\Gamma}:\mathcal{C}^{\Gamma}\rightleftarrows \mathcal{D}^{\Gamma}:U^{\Gamma}\]
with respect to either the injective or the projective model structures.
\end{lem}

\begin{lem}\label{lem:deltaLQ}
Let $\cte:\mathrm{Sp}^\mathcal{O}_\mathrm{proj}\to B(\mathcal{O}, \mathrm{Sp})$ be the functor defined in \eqref{eq:defiDelta}. Then $\cte$ is a left Quillen functor.
\end{lem}
\begin{proof}
Let $\operatorname{const}:\mathrm{Sp}\to\mathrm{Sp}^{\mathcal{O}^\mathit{op}}_\mathrm{inj}$ be the functor that sends a spectrum $E$ to the constant functor on $E$. Then $\operatorname{const}$ is left Quillen since it clearly sends cofibrations (resp. trivial cofibrations) to cofibrations (resp. trivial cofibrations) and it has a right adjoint (taking limit). Note that we have:
\[\cte=(\operatorname{const})^\mathcal{O}:\mathrm{Sp}^{\mathcal{O}}_\mathrm{proj}\to \left[\mathrm{Sp}^{\mathcal{O}^\mathit{op}}_\mathrm{inj}\right]^\mathcal{O}_\mathrm{proj}=B(\mathcal{O},\mathrm{Sp})\]
Then $\cte$ is left Quillen by Lemma \ref{lem:htt}.
\end{proof}

\begin{lem}\label{lem:restIsLeftQuillen}
Let $\mathcal{C}$ be a combinatorial model category, $\Gamma$ be a small category, fix $t\in\Gamma$, and let $u_t:\slice{\Gamma}{t}\to\Gamma$ be the forgetful functor from the slice category. Then the restriction on both variables
\[\Ustar{t}:B(\Gamma,\mathcal{C})\toB(\slice{\Gamma}{t}, \mathcal{C})\]
is a left Quillen functor.
\end{lem}
\begin{proof}
Consider the following commutative square:
\[\xymatrix{B(\Gamma,\mathcal{C})=(\mathcal{C}^{\Gamma^\mathit{op}}_\mathrm{inj})^{\Gamma}_\mathrm{proj}\ar[dr]^-{\Ustar{t}}\ar[r]^-{\Uast{t}}\ar[d]_-{\Uast{t}} & (\mathcal{C}^{\Gamma^\mathit{op}}_\mathrm{inj})^{\slice{\Gamma}{t}}_\mathrm{proj} \ar[d]^-{\Uast{t}} \\
\left[\mathcal{C}^{(\slice{\Gamma}{t})^\mathit{op}}_\mathrm{inj}\right]^{\Gamma}_\mathrm{proj} \ar[r]^-{\Uast{t}} & \left[\mathcal{C}^{(\slice{\Gamma}{t})^\mathit{op}}_\mathrm{inj}\right]^{\slice{\Gamma}{t}}_\mathrm{proj}=B(\slice{\Gamma}{t}, \mathcal{C})}\]
The horizontal morphisms are left Quillen by Lemma \ref{lem:resLeftQuillen2} \eqref{item:adj1} and the vertical morphisms are left Quillen by Lemma \ref{lem:resLeftQuillen1} and Lemma \ref{lem:htt}. Then $\Ustar{t}$ is left Quillen as well, for being the composite of left Quillen functors.
\end{proof}



\subsection{The natural zig-zag}\label{sec:zznat}
In this section we finally construct a zig-zag of $\mathcal{O}$-spectra inducing \eqref{eq:natAdjGroups} upon taking homotopy groups. We begin with the following lemma, that shows that every $\mathcal{O}$-spectrum can be canonically described as an objectwise coend.
\begin{lem}\label{lem:nat:isofun}
Let $\cte:\mathrm{Sp}^\mathcal{O}_\mathrm{proj}\to B(\mathcal{O}, \mathrm{Sp})$ be the functor defined in \eqref{eq:defiDelta}. For $F\in\mathrm{Sp}^\mathcal{O}$ and $t\in\mathcal{O}$, the structural morphisms into the coends
\begin{equation}\label{eq:nat:isofun}
\xymatrix{F(t)=\left[\Ustar{t}\cte[F]\right](\mathrm{id}_t,\mathrm{id}_t)\ar[r] & \textstyle\intg{t}\Ustar{t}\cte[F]}
\end{equation}
are isomorphisms. Moreover, these are natural in $t\in\mathcal{O}$ and in $F\in\mathrm{Sp}^\mathcal{O}$.
\end{lem}
\begin{proof}
Fix $t\in\mathcal{O}$. Since $\Ustar{t}\cte[F]$ is constant in the contravariant variable, we have:
\[\textstyle\int^{\slice{\mathcal{O}}{t}}\Ustar{t}\cte[F]\cong \displaystyle\operatornamewithlimits{colim}_{\alpha\in\slice{\mathcal{O}}{t}}F(u_t(\alpha))\]
Since $\mathrm{id}_t$ is a final object of $\slice{\mathcal{O}}{t}$, the structural morphism
\[\xymatrix{F(t)=F(u_t(\mathrm{id}_t))\ar[r] & \displaystyle\operatornamewithlimits{colim}_{\alpha\in\slice{\mathcal{O}}{t}}F(u_t(\alpha))}\]
is an isomorphism. Combining the above we get the desired isomorphism:
\[\xymatrix{F(t)\ar[r]^-{\cong} & \textstyle\int^{\slice{\mathcal{O}}{t}}\Ustar{t}\cte[F]}\]
It is easily verified that this is natural in $t$ and in $F$.
\end{proof}


\begin{lem}\label{lem:natUnit}
Let $F\in\mathrm{Sp}^\mathcal{O}$ and fix $t\in\mathcal{O}$. Let $\Uexcl{t}$ and $\Uast{t}$ be the functors that form the adjunction \eqref{eq:adjResPF}. Then there is a morphism of bifunctors $\Ustar{t}\cte[F]\to \Ustar{t}\Uexcl{t}\Uast{t}\cte[F]$
described as follows. For objects $\alpha:r\to t$ and $\beta:s\to t$ of $\slice{\mathcal{O}}{t}$, the component corresponding to the pair $(\beta,\alpha)$ is the structural morphism into the coproduct corresponding to $\beta$:
\begin{equation}\label{eq:lemNatUnit}\xymatrix{
\left[\Ustar{t}\cte[F]\right](\beta,\alpha)=F(r)\ar[r]^-{\mathit{can}_\beta} &
\displaystyle\coprod_{\mathcal{O}(s,t)}F(r)=\left[\Ustar{t}\Uexcl{t}\Uast{t}\cte[F]\right](\beta, \alpha)
}\end{equation}
Moreover, upon taking coend we get a morphism of spectra
\begin{equation}\label{eq:nat:unit}\xymatrix{
\intg{t}\Ustar{t}\cte[F]\ar[r] & \intg{t}\Ustar{t}\Uexcl{t}\Uast{t}\cte[F]
}\end{equation}
that is natural in $t$.
\end{lem}
\begin{proof}
The fact that the morphisms \eqref{eq:lemNatUnit} are natural in $\alpha$ and $\beta$ is easily verified using the explicit description of the bifunctor $\Uexcl{t}\Uast{t}\cte[F]$ given in Remark \ref{rem:constr}.

Let us now prove that \eqref{eq:nat:unit} is natural in $t$. Let $h:t\to t'$ be a morphism in $\mathcal{O}$. By the universal property of the coend, it suffices to show that the outer square in the following diagram commutes for every $\alpha:r\to t$ in $\slice{\mathcal{O}}{t}$:
\[\xymatrix@C=2em@R=2em{F(r)=[\Ustar{t}\cte[F]](\alpha, \alpha)\ar[dr]^-{\mathit{can}_\alpha}\ar@/^1pc/[rrd]\ar@/_2pc/[ddr] & & \\
& \intg{t}\Ustar{t}\cte[F]\ar[d]_-{h_*}\ar[r]^-{\eqref{eq:nat:unit}} &
\intg{t}\Ustar{t}\Uexcl{t}\Uast{t}\cte[F]\ar[d]^-{h_*} \\
& \intg{t'}\Ustar{t'}\cte[F]\ar[r]^-{\eqref{eq:nat:unit}} &
\intg{t'}\Ustar{t'}\Uexcl{t'}\Uast{t'}\cte[F]
}\]
Unravelling the definitions of $h_*$ (see Lemma \ref{lem:nat} and Remark \ref{rem:constr}), it is straightforward to verify that both ways from $F(r)$ to $\intg{t'}\Ustar{t'}\Uexcl{t'}\Uast{t'}\cte[F]$ in the diagram above equal the composite:
\[\xymatrix@C=4em{F(r)\ar[r]^-{\mathit{can}_{h\circ\alpha}} &
\displaystyle\coprod_{\mathcal{O}(r, t')}F(r)\ar[r]^-{\mathit{can}_{h\circ\alpha}} &
\intg{t'}\Ustar{t'}\Uexcl{t'}\Uast{t'}\cte[F]
}\]
This finishes the proof.
\end{proof}

\begin{thm}\label{thm:ZZAG}
Let $q:Q\xrightarrow{\sim} \mathrm{id}$ and $\bar{q}:\bar{Q}\xrightarrow{\sim}\mathrm{id}$ be, respectively, cofibrant replacements in $B(\mathcal{O},\mathrm{Sp})$ and $\mathrm{Sp}^{\mathcal{O}}_{\mathrm{proj}}$. Then we have a zig-zag of $\mathcal{O}$-spectra as follows:
\[
\xymatrix@C=1.6em{
\intg{t} \Ustar{t} \cte[J] &
\intg{t} \Ustar{t} \cte[\bar{Q} J]\ar[l]^-{\sim}_-{\bar{q}}\ar[r]^-{\eqref{eq:nat:unit}} &
\intg{t} \Ustar{t} \Uexcl{t} \Uast{t} \cte[\bar{Q}J] \\
\intg{t} \Ustar{t} Q (M_t) &
\intg{t} \Ustar{t} Q \Uexcl{t} \Uast{t} \cte[J]\ar[l]_-{\varphi}  &
\intg{t} \Ustar{t} Q \Uexcl{t} \Uast{t} \cte[\bar{Q}J]\ar[u]_-{q}^-{\rotatebox{90}{$\scriptstyle\sim$}}\ar[l]_-{\bar{q}} \\
\intg{t} \Ustar{t} Q (L_t )\ar[r]^-{q}\ar[u]^-{\psi}_-{\rotatebox{90}{$\scriptstyle\sim$}} &
\intg{t} \Ustar{t} L_t \ar[r]^-{\zeta} &
\mathbb{K}(A^{(t)}, M_GB)
}
\]
Moreover, upon identifying
\[\mathbb{K}(A, \mathcal{R}(B\rtimes t))=J(t)\overset{\eqref{eq:nat:isofun}}{\cong} \intg{t}\Ustar{t}\Const{J}\]
and then taking homotopy groups, this zig-zag induces the isomorphism \eqref{eq:natAdjGroups}.
\end{thm}
\begin{proof}
Let us first show that the morphisms appearing in the zig-zag are indeed natural in $t$. For \eqref{eq:nat:unit} this is part of Lemma \ref{lem:natUnit} and for $\zeta$ it is Lemma \ref{lem:natZeta}. The rest of the morphisms are natural in $t$ because they result from aplying the functor $\funcoend$ of Lemma \ref{lem:nat} to appropriate morphisms in $B(\mathcal{O}, \mathrm{Sp})^\mathcal{O}$.

The morphisms in the zig-zag labeled with $\sim$ are indeed weak equivalences by Ken Brown's Lemma \cite{hoveymodel}*{Lemma 1.1.12}: they result from applying a left Quillen functor to an appropriate weak equivalence between cofibrant objects. Here we use that the functors $\cte$, $\Ustar{t}$ and $\intg{t}$ are left Quillen by Lemmas \ref{lem:deltaLQ}, \ref{lem:restIsLeftQuillen} and Theorem \ref{thm:coendquillen} respectively.

The fact that the zig-zag induces \eqref{eq:natAdjGroups} upon taking homotopy groups follows from the commutativity of the following diagram of spectra.

\[
\xymatrix{
\framebox{$\mathbb{K}(A, \mathcal{R}(B\rtimes t))$}
\ar@{=}[r]
\ar[dd]^-{\mathit{can}_{\mathrm{id}_t}} &
J(t)
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}}_-{\text{$\cong$ by Lem. \ref{lem:nat:isofun}}} &
\intg{t}\Ustar{t}\cte[J] \\
& (\bar{Q}J)(t)
\ar[u]^-{\bar{q}}_-{\rotatebox{90}{$\scriptstyle\sim$}}
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}}_-{\text{$\cong$ by Lem. \ref{lem:nat:isofun}}}
\ar[d]_-{\mathit{can}_{\mathrm{id}_t}} &
\intg{t}\Ustar{t}\cte[\bar{Q}J]
\ar[u]^-{\bar{q}}_-{\rotatebox{90}{$\scriptstyle\sim$}}
\ar[d]_-{\eqref{eq:nat:unit}} \\
\displaystyle\coprod_{\gamma\in\mathcal{O}(t,t)}J(t)
\ar[ddd]^-{\varphi} &
\displaystyle\coprod_{\gamma\in\mathcal{O}(t,t)}(\bar{Q}J)(t)
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}}
\ar[l]_-{\bar{q}} &
\intg{t}\Ustar{t}\Uexcl{t}\Uast{t}\cte[\bar{Q}J] \\
& \left[Q\Uexcl{t}\cte[\bar{Q}J]\right](t,t)
\ar[u]^-{q}_-{\rotatebox{90}{$\scriptstyle\sim$}}
\ar[d]_-{\bar{q}}
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}} &
\intg{t}\Ustar{t}Q\Uexcl{t}\Uast{t}\cte[\bar{Q}J]
\ar[u]^-{q}_-{\rotatebox{90}{$\scriptstyle\sim$}}
\ar[d]_-{\bar{q}} \\
& \left[Q\Uexcl{t}\cte[J]\right](t,t)
\ar[d]_-{\varphi}
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}} &
\intg{t}\Ustar{t}Q\Uexcl{t}\Uast{t}\cte[J]
\ar[d]_-{\varphi} \\
M_t(t,t) & (QM_t)(t,t)
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}}
\ar[l]_-{q}^-{\sim} &
\intg{t}\Ustar{t}Q(M_t) \\
& (QL_t )(t,t)
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}}
\ar[u]^-{\psi}_-{\rotatebox{90}{$\scriptstyle\sim$}}
\ar[d]_-{q} &
\intg{t}\Ustar{t}Q(L_t )
\ar[u]^-{\psi}_-{\rotatebox{90}{$\scriptstyle\sim$}}
\ar[d]_-{q} \\
& L_t (t,t)
\ar[r]^-{\mathit{can}_{\mathrm{id}_t}}
\ar@/^2pc/[uul]^-{\psi}_-{\hspace{-0.5em}\rotatebox{135}{$\scriptstyle\sim$}}
\ar@/_1pc/[dr]_-{\zeta_t} &
\intg{t}\Ustar{t}L_t 
\ar[d]^-{\zeta} \\
& & \framebox{$\mathbb{K}^G(A^{(t)}, M_GB$)}
\ar@(l,dr) "5,2";"3,1"^-{q}_-{\hspace{-0.5em}\rotatebox{135}{$\scriptstyle\sim$}}
\ar@(l,l)@/_3pc/ "1,1";"6,1"_-{(-)^{(t)}}
}
\]
Upon taking homotopy groups, the morphisms labeled with $\sim$ in the diagram above become isomorphisms and can be inverted. The zig-zag equals the morphism in the top row followed by the composite of the morphisms in the rightmost column. By Lemma \ref{lem:natcounit} \eqref{item:couniadj}, the composite of the bent morphisms on the left equals, upon taking homotopy groups, the isomorphism \eqref{eq:natAdjGroups}.
\end{proof}


\section{Equivariant homology with coefficients in \texorpdfstring{$kk^G$}{kkG}-theory}\label{sec:prin}

\begin{lem}\label{lemaprevio}
Let $A$ be an algebra, $B$ be a $G$-algebra and $X$ be a $(G, \FIN)$-complex. Then there exists a morphism of spectra
\begin{equation}\label{eq:transNat}
\alpha_{X}: H^G(X; \mathbb{K}^G(A^{(-)}, B))\rightarrow 
\mathbb{K}^G(A^{(X)}, B)\end{equation}
that is natural in $X$ and that is an isomorphism for $X=G/H$.
\end{lem}
\begin{proof}
Let $\bold{E}:\OrFin{G}\to \mathrm{Sp}$ be a functor and $X$ be a $(G, \FIN)$-complex. By Lemma \ref{lem:fixcolim}, Lemma \ref{lem:prodCat} and the fact that coends and smash products commute with colimits, we have:
\begin{align*}
    \framebox{ $H^G(X; \bold{E})$}&= \int^{G/H}X^H_+\wedge \bold{E}(G/H) \\ \\
    &\cong \int^{G/H} \left(\operatornamewithlimits{colim}_{G/K\times \Delta^n\downarrow X} G/K\times \Delta^n\right)^H_+ \wedge \bold{E}(G/H) \\ \\
    &\cong \int^{G/H} \operatornamewithlimits{colim}_{G/K\times \Delta^n\downarrow X} (G/K\times\Delta^n)^H_+ \wedge \bold{E}(G/H) \\ \\
    &\cong \operatornamewithlimits{colim}_{G/K\times \Delta^n\downarrow X} \Delta^n_+ \wedge \int^{G/H}  (G/K)^H_+ \wedge \bold{E}(G/H) \\ \\
    &\cong \operatornamewithlimits{colim}_{G/K\times \Delta^n\downarrow X} \Delta^n_+ \wedge H^G(G/K; \bold{E}) \\ \\
    &\cong \framebox{$\displaystyle\operatornamewithlimits{colim}_{G/K\times \Delta^n\downarrow X} \Delta^n_+ \wedge \bold{E}(G/K)$}
\end{align*}
Thus, to construct the morphism \eqref{eq:transNat} it suffices to define compatible morphisms \begin{equation}\label{eq:compatMorph}\Delta^n_+\wedge \mathbb{K}^G(A^{(G/K)},B)\to \mathbb{K}^G(A^{(X)},B)\end{equation}
for every $f:G/K\times \Delta^n\to X$. Define \eqref{eq:compatMorph} as the composite:
\[\xymatrix{\Delta^n_+\wedge\mathbb{K}^G(A^{(G/K)},B)\ar[r]^-{\eqref{eq:lemMult}} &
\mathbb{K}^G(A^{(G/K\times \Delta^n)},B)\ar[r]^-{f_*} &
\mathbb{K}^G(A^{(X)},B)}\]
The compatibility of these morphisms is immediate from the naturality of \eqref{eq:lemMult} in $G/K$. To see that the induced morphism \eqref{eq:transNat} is an isomorphism for $X=G/H$, note that in this case the identity of $X\cong G/H\times \Delta^0$ is a final object among those $G/K\times\Delta^n\downarrow X$. Then, in this case, taking colimit over $G/K\times\Delta^n\downarrow X$ boils down to evaluating at the final object $\mathrm{id}_{G/H\times \Delta^0}$ and the result follows from Lemma \ref{lem:mult}.
\end{proof}

\begin{lem}\label{pruebaparte1}
Let $A$ be an algebra and $B$ be a $G$-algebra. For every $(G, \FIN)$-complex $Y$ there is a natural weak equivalence of spectra:
\[H^G(Y; \mathbb{K}^G(A^{(-)}, B))\xrightarrow{\sim} \operatornamewithlimits{colim}_{\substack{X\subseteq Y\\ \text{$(G, \FIN)$-finite}}} \mathbb{K}^G(A^{(X)}, B)\]
\end{lem}
\begin{proof}
For every $(G, \FIN)$-complex $Y$ we have that 
\begin{equation}\label{xcoly}
Y\cong \operatornamewithlimits{colim}_{\substack{X\subseteq Y\\ \text{$(G, \FIN)$-finite}}} X. 
\end{equation}
From Lemma \ref{lemaprevio} there exist morphims of spectra  $\alpha_{X}$
that are natural in $X$. Since homology commutes with filtered colimits, we have a morphism of spectra induced by the $\alpha_X$:
\[
\displaystyle{
\beta_{Y}=\operatornamewithlimits{colim}{\alpha_{X}}:  H^G(Y; \mathbb{K}^G(A^{(-)}, B))  \rightarrow  \displaystyle{\operatornamewithlimits{colim}_{\substack{X\subseteq Y\\  \text{$(G, \FIN)$-finite}}}} \mathbb{K}^G(A^{(X)}, B)
}
\]
Let us prove that the morphism $\beta_{Y}$ is a weak equivalence. It suffices to show that $\alpha_X$ is a weak equivalence for every $(G, \FIN)$-finite $X$. 
We are going to prove the latter following these steps:
\begin{itemize}
\item[(a)] $X=G/H$ for $H\in \mathcal{F}in$
\item[(b)] $X=G/H \times \Delta^{n}$ for $H\in \mathcal{F}in$ and $n\in\mathbb{N}$
\item[(c)] $\displaystyle{X=\bigsqcup^{m}_{i=1} G/H_{i}\times \Delta^{n} }$ for $H_{i}\in \mathcal{F}in$ and $m,n\in\mathbb{N}$
\item[(d)] $X$ any $(G, \FIN)$-finite complex
\end{itemize}
The first step is proved in Lemma \ref{lemaprevio}.
If $X=G/H\times \Delta^{n}$ we obtain the following diagram:
$$
\xymatrix{ H^G(G/H\times \Delta^{n}; \mathbb{K}^G(A^{(-)}, B)) \ar[r] \ar[d]_{\pi_{*}}& \mathbb{K}^G(A^{(G/H\times \Delta^{n})}, B) \ar[d]^{\pi_{*}}\\
H^G(G/H; \mathbb{K}^G(A^{(-)}, B)) \ar[r] & \mathbb{K}^G(A^{(G/H)}, B)
}
$$
The left vertical arrow is an equivalence by homotopy invariance of the equivariant homology theory. The right vertical arrow is a weak equivalence because $A^{(G/H\times \Delta^{n})}\cong A^{(G/H)}\otimes A^{\Delta^{n}}$ and by homotopy invariance of $kk_{*}^{G}(-,B)$.

Consider $\displaystyle{X=\bigsqcup^{m}_{i=1} G/H_{i}\times \Delta^{n}}$ and write $\bold{E} = \mathbb{K}^G(A^{(-)}, B)$. Note that this step follows from the previous one because:
\begin{align*}
H^{G}(X;\bold{E})&\cong \bigoplus^{n}_{i=1}H^{G}_{*}(G/H_{i} \times \Delta^{n};\bold{E})) \\
kk^{G}(A^{(X)}, B)&\cong  \bigoplus_{i=1}^{n}kk_{*}^{G}(A^{(G/H_{i}\times \Delta^{n})},B) \\
\alpha_X&=\oplus \alpha_{G/H_i\times \Delta^n}
\end{align*}
Let us assume that $\alpha_X$ is a weak equivalence for every $(G,\mathcal{F}in)$-finite $X$ complex of dimension $n-1$. Consider $X$ obtained as the following pushout diagram:
$$
\xymatrix{\displaystyle \bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n} \ar[r] \ar@{^{(}->}[d] & X_{n-1} \ar@{^{(}->}[d] & \\
\displaystyle \bigsqcup_{i=1}^{n}G/H_{i} \times \Delta^{n} \ar[r] & X
}
$$
By Lemma \ref{lem:milnor}, upon applying the functor $A^{(-)}$ we obtain the following Milnor square:

$$
\xymatrix{A^{(\bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n} )}  & A^{(X_{n-1})}  \ar[l]  \\
\displaystyle A^{(\bigsqcup_{i=1}^{n}G/H_{i} \times \Delta^{n})} \ar@{->>}[u] & A^{(X)} \ar@{->>}[u] \ar[l] 
}
$$
Then 
$$
\xymatrix{
\vdots \ar[d] & \vdots \ar[d] \\
H_{*}^{G}(\bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n}, \bold{E}) \ar[d]  \ar[r]^{\alpha_{\bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n}}} & kk_{*}^{G}(A^{(\bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n})}, B) \ar[d] \\
H_{*}^{G}(\bigsqcup_{i=1}^{n}G/H_{i} \times \Delta^{n}, \bold{E}) \oplus H_{*}^{G}(X_{n-1}, \bold{E})\ar[d] \ar[r]^{\underset{}{\alpha_{\bigsqcup_{i=1}^{n}G/H_{i} \times \Delta^{n}}} \oplus \alpha_{X_{n-1}}}& kk_{*}^{G}(A^{(\bigsqcup_{i=1}^{n}G/H_{i} \times\Delta^{n})}, B) \oplus kk^{G}_{*}(A^{(X_{n-1})},B)  \ar[d]  \\
H_{*}^{G}(X, \bold{E}) \ar[d] \ar[r]^{\alpha_{X}} & kk^{G}_{*}(A^{(X)},B) \ar[d] \\ 
H_{*-1}(\bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n}, \bold{E}) \ar[r] \ar[d]  &  kk_{*-1}^{G}(A^{(\bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n})},B ) \ar[d] \\
\vdots & \vdots }
$$
The left column is given by Mayer-Vietoris sequence for $H^{G}(-;\bold{E})$ and the right column is given by excision of $kk^{G}$; see Lemma \ref{lem:excision}. 
By induction, $\alpha_{\bigsqcup_{i=1}^{n}G/H_{i} \times \partial \Delta^{n}}$ and $\alpha_{X_{n-1}}$ are isomorphisms. 
By step (c), $\alpha_{\bigsqcup_{i=1}^{n}G/H_{i} \times \Delta^{n}}$ is an isomorphism. We conclude by the Five Lemma that $\alpha_{X}$ is an isomorphism. 
\end{proof}

\begin{thm}\label{mainthm}
Let $G$ be a group  satisfying \eqref{eq:hipG}. Let $A$ be an algebra, $B$ be a $G$-algebra and $Y$ be a $(G, \FIN)$-complex. Then there is a natural isomorphism
\[H^G_*(Y; \mathbb{K}(A, \mathcal{R}(B\rtimes -)))\cong \operatornamewithlimits{colim}_{\substack{X\subseteq Y\\\text{$(G, \FIN)$-finite}}} kk^G_*(A^{(X)}, B)\]
induced by a natural zig-zag of spectra.
\end{thm}
\begin{proof}
Define $\bold{E},\bold{F}:\OrFin{G}\to\mathrm{Sp}$ by:
\begin{align*}
\bold{E}(G/N) &= \mathbb{K}(A,\mathcal{R}(B\rtimes G/N)) \\
\bold{F}(G/N) &= \mathbb{K}^G(A^{(G/N)},B) 
\end{align*}
By Theorem \ref{thm:ZZAG}, there is a zig-zag of $\OrFin{G}$-spectra
\begin{equation}\label{eq:mainThmZzag}
    \xymatrix{ \bold{E} \ar@{-->}[r] & \bold{F} }
\end{equation}
that induces the isomorphism \eqref{eq:natAdjGroups} upon taking homotopy groups. Moreover, the morphisms in \eqref{eq:mainThmZzag} going backwards are objectwise weak equivalences of spectra. Let $Y$ be a $(G, \FIN)$-complex. After applying $H^G(Y;-)$ to \eqref{eq:mainThmZzag} we get a zig zag of spectra:
\begin{equation}\label{eq:mainThmZzag2}
    \xymatrix{ H^G(Y;\bold{E}) \ar@{-->}[r] & H^G(Y;\bold{F}) }
\end{equation}
Here the morphisms going backwards are weak equivalences by \cite{dalu}*{Lemma 4.6}. Upon taking homotopy groups, we get an homomorphism:
\begin{equation}\label{eq:mainThmTrans}
    \xymatrix{ H^G_*(Y;\bold{E}) \ar[r] & H^G_*(Y;\bold{F}) }
\end{equation}
This morphism is clearly natural in $Y$. Moreover, we claim that it is an isomorphism. The assertion holds for $Y=G/N$ since, in this case, it is the isomorphism \eqref{eq:natAdjGroups}. Now continue as in the proof of Lemma \ref{pruebaparte1}: the case $Y=G/H\times\Delta^n$ follows from homotopy invariance and the general case follows from excision considering the skeletal filtration of $Y$; see also \cite{dalu}*{Theorem 6.3 (2)}. Finally, combine \eqref{eq:mainThmTrans} with Lemma \ref{pruebaparte1}:
\[\xymatrix@C=2em{
H^G_*(Y; \mathbb{K}(A, \mathcal{R}(B\rtimes -)))\ar[r]^-{\eqref{eq:mainThmTrans}}_-{\cong} &
H^G_*(Y; \mathbb{K}^G(A^{(-)}, B)\ar[r]_-{\cong} &
\displaystyle\operatornamewithlimits{colim}_{\substack{X\subseteq Y\\\text{$(G, \FIN)$-finite}}} kk^G_*(A^{(X)}, B)
}\]
This finishes the proof.
\end{proof}

