Given a continuous map of topological spaces $f: X \to Y$ there is a description of the higher direct images saying that $R^i f_* \mathcal{F}$ is the sheaf on $Y$ associated to the presheaf $$ U \mapsto H^i(f^{-1}(U), \mathcal{F}). $$
My question is whether a similar description is true for higher direct images with proper support (under adequate assumptions for $X$ and $Y$), namely that $R^i f_! \mathcal{F}$ is the sheaf on Y associated to the presheaf $$ U \mapsto H^i_c(f^{-1}(U), \mathcal{F}). $$
The proof of the former that I know does not apply, since $f_!$ does not decompose as an exact functor of presheaves followed by sheafification. I have consulted Kashiwara-Schapira but I could not find anything on this regard, so any reference would be greatly appreciated.