Right derived functors of composite

Question

I think I have proven the following.

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories such that $\mathcal{A}$ and $\mathcal{B}$ have enough injectives. Let $F$ be an exact functor from $\mathcal{A}$ to $\mathcal{B}$ which maps injectives to injectives. Let $G$ be a left exact functor from $\mathcal{B}$ to $\mathcal{C}$. Then the composite $G \circ F$ is again left exact, and the right derived functors satisfy $$ \operatorname{R}^n(G \circ F) \cong \operatorname{R}^n(G) \circ F \,. $$

I’m looking for a reference (a book or an article) which confirms this result.

Edit. I have no experience with spectral sequences, so I’m looking for a sufficiently basic reference.

Answer 1

If $F$ is only assumed to be left exact, the composite is still left exact and the right derived functors can be computed with the Grothendieck spectral sequence. That $F$ is exact is a degenerate case, which might be treated by any author treating the general case. In any case your result follows immediately from the Grothendieck SS. I learned about it from Rotman an introduction to homological algebra.