Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories and let $G: \mathcal{A} \to \mathcal{B}, F: \mathcal{B} \to \mathcal{C}$ be left exact functors, with the hypotheses needed to apply the Grothendieck spectral sequence. The edge morphisms of said spectral sequence give us natural maps $$ R^p F \circ G \to R^p(F \circ G) \quad \text{and} \quad R^q(F \circ G) \to F \circ R^q G.$$
If $F$ is actually exact, one can see directly from the definition using injective resolutions that there is an isomorphism $R^q(F \circ G) \cong F \circ R^q G$, so the first question would be:
- Is the edge map on the right (of the two above) said isomorphism?
On the other hand, if $G$ is exact I have only managed to construct a natural map $R^p(F \circ G) \to R^p F \circ G$ using injective resolutions. However, here comes the second question:
- How does this map relate to the edge map on the left (of the two above)? Are they inverses? If not, what can we say about their compositions?
I don't know how to relate both approaches to the maps (edge maps and injective resolutions) so I have no idea how to compare them.
