Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\bullet\to A\to 0$ is a resolution of $A$ by $F$-acyclic objects, then $L_nF(A)\cong H_n(Q_\bullet)$. Suppose now that $f\colon A\to A'$ is a morphism in $\mathcal A$ and $Q'_\bullet \to A'\to 0$ is a resolution of $A'$ by $F$-acyclic objects. Suppose moreover, I have a chain map $Q_\bullet\to Q'_\bullet$ over $f$. Then I get induced homomorphism $f_n\colon H_n(Q_\bullet)\to H_n(Q'_\bullet)$ which agree with $f$ at the level of $H_0$ (after the natural identifications). My question is can we choose isomorphisms $L_nF(A)\to H_n(Q_\bullet)$ and $L_nF(A')\to H_n(Q'_\bullet)$ so that these isomorphisms intertwine $f_n$ and $L_nF(f)$ for all $n$?

Any pointers to references would be appreciated.

Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\bullet\to A\to 0$ is a resolution of $A$ by $F$-acyclic objects, then $L_nF(A)\cong H_n(F(Q_\bullet))$. Suppose now that $f\colon A\to A'$ is a morphism in $\mathcal A$ and $Q'_\bullet \to A'\to 0$ is a resolution of $A'$ by $F$-acyclic objects. Suppose moreover, I have a chain map $Q_\bullet\to Q'_\bullet$ over $f$. Then I get induced homomorphism $f_n\colon H_n(F(Q_\bullet))\to H_n(F(Q'_\bullet))$ which agree with $f$ at the level of $H_0$ (after the natural identifications). My question is can we choose isomorphisms $L_nF(A)\to H_n(F(Q_\bullet))$ and $L_nF(A')\to H_n(F(Q'_\bullet))$ so that these isomorphisms intertwine $f_n$ and $L_nF(f)$ for all $n$?

Any pointers to references would be appreciated.