The Stacks project

Lemma 13.30.2. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{A}$ be functors of abelian categories such that $F$ is a right adjoint to $G$. Let $K^\bullet $ be a complex of $\mathcal{A}$ and let $M^\bullet $ be a complex of $\mathcal{B}$. If $RF$ is defined at $K^\bullet $ and $LG$ is defined at $M^\bullet $, then there is a canonical isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{B})}(M^\bullet , RF(K^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(LG(M^\bullet ), K^\bullet ) \]

This isomorphism is functorial in both variables on the triangulated subcategories of $D(\mathcal{A})$ and $D(\mathcal{B})$ where $RF$ and $LG$ are defined.

Proof. This is a special case of the very general Lemma 13.30.1. $\square$

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