Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss.
So all proofs I can find factors through a particular statement, which goes to Kashiwara-Shapira, which leaves it to readers as exercise ((2.6.25) on page 114).
The statement: Let $f:X\rightarrow Y$ be a map of ``nice'' spaces (finite CW-complex or even manifolds if you like, so every cohomological dimension ever appear is finite). Let $\mathcal{F},\mathcal{G}\in D^b(Sh(X))$ be two bounded complexes of sheaves (of abelian groups) on $X$. The statement is question asserts that we have a map
$$Rf_*R\underline{Hom}_X(\mathcal{F},\mathcal{G})\rightarrow R\underline{Hom}_Y(Rf_!\mathcal{F},Rf_!\mathcal{G})$$
that is functorial in $\mathcal{F}$ and $\mathcal{G}$. Here $\underline{Hom}$ is the sheaf Hom and $R\underline{Hom}$ its derived functor. This map (or rather natural transformation) is supposed to be derived from the natural map for sheaves
$$f_*\underline{Hom}_X(\mathcal{F},\mathcal{G})\rightarrow \underline{Hom}_Y(f_!\mathcal{F},f_!\mathcal{G})$$
which is fairly straightforward. But how do we derive it? In the $f_*$ version we can use that $\underline{Hom}_Y(f_*\mathcal{F},f_*\mathcal{G})$ is flabby for some injective $\mathcal{G}$. I can't see the same for $f_!$ ...
