Lemma 13.22.1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be left exact functors. Assume $\mathcal{A}$, $\mathcal{B}$ have enough injectives. The following are equivalent

1. $F(I)$ is right acyclic for $G$ for each injective object $I$ of $\mathcal{A}$, and

2. the canonical map

$t : R(G \circ F) \longrightarrow RG \circ RF.$

is isomorphism of functors of functors from $D^{+}(\mathcal{A})$ to $D^{+}(\mathcal{C})$.

Proof. If (2) holds, then (1) follows by evaluating the isomorphism $t$ on $RF(I) = F(I)$. Conversely, assume (1) holds. Let $A^\bullet$ be a bounded below complex of $\mathcal{A}$. Choose an injective resolution $A^\bullet \to I^\bullet$. The map $t$ is given (see proof of Lemma 13.15.16) by the maps

$R(G \circ F)(A^\bullet ) = (G \circ F)(I^\bullet ) = G(F(I^\bullet ))) \to RG(F(I^\bullet )) = RG(RF(A^\bullet ))$

where the arrow is an isomorphism by Lemma 13.17.7. $\square$

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