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

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.