I am wondering about the 2-functoriality in equivariant derived categories, and I hope that someone can clarify... (apologies if this is a stupid question)

For the more precise formulation, recall that the book of Bernstein-Lunts provides several definitions of equivariant derived categories $D^b_G(X)$ when $X$ is a $G$-variety. To define $f_\ast$ for an equivariant map $f:X\to Y$, we choose a skeleton of the category of resolutions of $Y$ containing smooth resolutions of arbitrarily high acyclicity. Via fiber product, these give resolutions of $X$ of arbitrarily high acyclicity and a morphism of diagrams along which we can push-forward sheaves. Similar things happen for the other functors $f_!$ and $f^!$.

The assignment $(G,X)\mapsto D^b_G(X)$ takes a $G$-scheme $X$ to the equivariant derived category. I would like to know if $f_\ast$, $f^\ast$, $f_!$ and $f^!$ are weak 2-functors from schemes with group action to triangulated categories. This seems problematic to me because of the choices made in the definition of the functors. These choices seem to imply that there is no coherent assignment of comparison isomorphism between $f_\ast\circ g_\ast$ and $(g\circ f)_\ast$.

So the questions are: is this an issue (maybe I am just missing a simple explanation)? How is it fixed? Can it be circumvented by some general higher category theory coherence theorems? Or can it be ignored for all things one wants to do with equivariant derived categories? Is it discussed somewhere in the literature on equivariant derived categories? (I did not find anything in the book of Bernstein-Lunts or anywhere else)


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