Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to FB \to FC$ is exact.
We know that for any right (resp. left) exact functor $F$, $L_nF$ (resp. $R^nF$) are middle exact, since they fit into a long exact sequence $$ ... \to L_n(A) \to L_n(B) \to L_n(C) \to ...$$
Is it then true that any middle exact functor $F$ comes from this construction, i.e. $F = L_nG$ or $R^nG$ for some $G$?
Is there a way to compute $G$ and $n$, given that we know $F=L_nG$ (or $R^nG$)?