For a map of topological spaces $j: X \to Y$, when do we have $j^{-1}RHom(\mathcal{F}^{\circ}, \mathcal{G}^{\circ}) \cong RHom(j^{-1}\mathcal{F}^{\circ},j^{-1}\mathcal{G}^{\circ})$? All constructions are taking place in $X$ and $Y$'s respective derived categories of sheaves. Although I'm not sure, I suspect one can answer the question for ordinary sheaves and then extrapolate to the derived setting.
Note this is true for $\mathcal{F}$ finitely presented and $X$ a point, since in that case $j^{-1}$ is just the stalk functor.