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

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

the canonical map

\[ t : R(G \circ F) \longrightarrow RG \circ RF. \]is isomorphism of functors of functors from $D^{+}(\mathcal{A})$ to $D^{+}(\mathcal{C})$.

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