Let $F$ be a left-exact functor from an abelian category $A$ to an abelian category $B$. Let $A'$ be a subcategory of $A$ (not necessarily abelian) and $B'$ a subcategory of $B$ which is abelian such that $R^iF(X)$ is an object of $B'$, for any object of $A'$. I am looking for a way to associate to objects of $A'$ some object in the derived category of $B'$ in a natural way. As a possible candidate, I suggest $RF(X)$, where $X$ will be an object of $A'$. Under what hypothesis on $F$ (and/or $A$,$A'$,$B$,$B'$) such assertion will hold? If you can find any other natural way to construct elements in $D(B')$ (starting from the data $(F,A,A’,B,B’)$, I am also interested.
Let me give one counter-example:
Let $A=\mathrm{Ab}$, the category of abelian group, $A'=A$ subcategory of $\mathrm{Ab}$ containing the object $\mathbb{Z}/p^2$. $B = B' = $ the category of $\mathbb{F}_p$-vector spaces, and $F = \ker$ of the multiplication by $p$. Then, $R^0F = F$ and $R^1F = \operatorname{coker}$ of the multiplication by $p$ have their image in $B$. However, $RF(\mathbb{Z}/p^2)=[\mathbb{Z}/p^2 \dashrightarrow \mathbb{Z}/p^2]$ is not in $D(B)$.