Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the equivalence conditions. Let $F:A \rightarrow D$ and $G: B \rightarrow D$ be left exact functors, I can choose an object $X$ in $A$, put it in an injective resolution
$0 \rightarrow X \rightarrow I^0 \rightarrow I^1 \rightarrow I^2 \rightarrow I^3 \rightarrow \cdots$
and compute the derived functors $R^nF(X)$, does the equivalance of categories help at all compute $R^nGP(X)$? Is there something to be gained from knowing $A$ and $B$ are equivalent in terms of computing derived functors?