I have the following problem. Let $\Gamma\_{G\_1\times G\_2}$ be a full subcategory of the orbit category $\mathcal{O}\_{G\_1\times G\_2}$ consisting of graph subgroups of $G\_1\times G\_2$. Further, let $N$ be a dual (i.e., covariant) coefficient system over $G\_1$ and define the dual coefficient system $FN$ over $G\_1\times G\_2$ to be:
$
FN(\Delta)=N(\pi\Delta)
$
if $\Delta$ is a graph subgroup and $0$ otherwise. Here $\pi\colon G\_1\times G\_2\to G\_1$ is a projection on the first factor.


Let $X$ be a $G\_1\times G\_2$-space which is $1\times G\_2$-free. I would like to show that
$$
H^{G\_1\times G\_2}\_\ast(X,FN)\cong H^{G\_1}\_\ast(\frac{X}{1\times G\_2},N).
$$


It's easy to see that this is true on $G\_1\times G\_2$-orbits (at least under graph subgroups) and I am pretty sure that there is some classical argument which can be used to prove this for general $G\_1\times G\_2$-spaces which are $1\times G\_2$-free, but I fail to see it. It seems to be fairly similar to the argument that the ordinary homology (non-equivariant) with given coefficients is unique and I remember that there is some spectral sequence argument for this proof - so also a reference for this fact would be very useful.



Reduction to graph subgroups for Bredon homology when the $G_1\times G_2$ is $G_2$-free
**TL; DR**: Suppose you have a functor between small categories $\mathcal C\_0\to \mathcal C$. Let $\mathcal D$ be a locally small category such as Top or Ch. Then there is an adjunction of functor categories
$$
L:[\mathcal C\_0, \mathcal D]\leftrightarrows [\mathcal C, \mathcal D]:R$$
Where the right adjoint $R$ is the restriction and $L$ is the left Kan extension. There is a similar adjunction between categories of contravariant functors. Now suppose we have functors $F\colon \mathcal C\_0\to \mathcal D$ and $G\colon 
\mathcal C^{\operatorname{op}}\to \mathcal D$. Then there is an isomorphism of coends:
$$ F\otimes\_{\mathcal C\_0} RG\cong LF \otimes\_{\mathcal C} G.$$


The isomorphism you ask about is an example of this adjunction.


------ Now with more details -------


By Elmendorf-type theorem, you can identify the category of $G\_1\times G\_2$-spaces that are $G\_2$-free with the category of functors $[\Gamma\_{G\_1\times G\_2}^{\operatorname{op}}, \mbox{Top}]$, where Top is the category of pointed spaces. Similarly, you can identify coefficient systems of the kind you consider with functors $[\Gamma\_{G\_1\times G\_2}, \mbox{Ch}]$, where Ch denotes the category of chain complexes. More precisely, coefficient systems correspond to functors that take values in chain complexes concentrated in degree zero.


The group homomorphism $G\_1\times G\_2\to G\_1$ induces a quotient by $G\_2$ functor $\Gamma\_{G\_1\times G\_2}\to \mathcal O\_{G\_1}$.


This functor induces an adjunction of functor categories
$$[\Gamma\_{G\_1\times G\_2}^{\operatorname{op}}, \mbox{Top}] \leftrightarrows [\mathcal O\_{G\_1}^{\operatorname{op}}, \mbox{Top}]$$
where the right adjoint is the pullback. The left adjoint, a.k.a the left Kan extension, can be identified, once again, with the quotient by $G\_2$ functor $X\mapsto \frac{X}{1\times G\_2}$. Here I have identified the functor categories with spaces with action of $G\_1\times G\_2$ and of $G\_1$ respectively.


Similarly there is a pair of adjoint functors between categories of coefficient systems


$$[\Gamma\_{G\_1\times G\_2}, \mbox{Ch}] \leftrightarrows [\mathcal O\_{G\_1}, \mbox{Ch}].$$
Here, again, the right adjoint is the pullback, and it is equivalent to the functor $F$ that you describe.


Finally, the Bredon homology groups $H\_\*^{G\_1\times G\_2}(X, FN)$ can be identified with the homology groups of the coend $$C\_\*\left(X^H\right)\otimes\_{H\in \Gamma\_{G\_1\times G\_2}} FN$$
and similarly the Bredon homology groups
$H\_\*^{G\_1}\left(\frac{X}{1\times G\_2}, N\right)$ can be identified with the homology groups of the following coend
$$C\_\*\left(\frac{X}{1\times G\_2}^H\right)\otimes\_{H\in \mathcal O\_{G\_1}} N.$$
The equivalence of the two coends follows from the adjunction.


