Our research group is currently going through the paper by Thomas Nikolaus and Peter Scholze On Topological Cyclic Homology and in the Appendix B, Proposition B.5., they use the notation $Fun^{B \mathbb Z}( \mathcal C, \mathcal D)$ for the $\infty-$category of $B \mathbb Z$-equivariant functors between two $\infty-$categories $\mathcal C, \mathcal D$ with $B \mathbb Z$ action. The notation hasn't been defined before, at least we couldn't find a definition, and a quick look at Lurie's HHT didn't give anything either.

My assumption is that for a simplicial group $G$ we can define $Fun^{G}( \mathcal C, \mathcal D)$ as a full subcategory of $Fun(BG,Fun( \mathcal C, \mathcal D))$. Two major properties that are used are:

- There is a natural map (isomorphism?) $ Fun ( C / G , \mathcal D) \rightarrow Fun^G ( C , \mathcal D) $ for $\mathcal D$ having a trivial $G$ action and
- $Fun^G ( pt , \mathcal D) \cong Fun ( BG , \mathcal D) $. This would follow from 1 if we do have an isomorphism and $Fun^G$ were invariant under homotopy equivalences, since $pt \simeq EG$ and $EG / G = BG$.