Lemma 13.28.4. 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. Since $RF$ is defined at $K^\bullet$, we see that the rule which assigns to a quasi-isomorphism $s : K^\bullet \to I^\bullet$ the object $F(I^\bullet )$ is essentially constant as an ind-object of $D(\mathcal{B})$ with value $RF(K^\bullet )$. Similarly, the rule which assigns to a quasi-isomorphism $t : P^\bullet \to M^\bullet$ the object $G(P^\bullet )$ is essentially constant as a pro-object of $D(\mathcal{A})$ with value $LG(M^\bullet )$. Thus we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{B})}(M^\bullet , RF(K^\bullet )) & = \mathop{\mathrm{colim}}\nolimits _{s : K^\bullet \to I^\bullet } \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{B})}(M^\bullet , F(I^\bullet )) \\ & = \mathop{\mathrm{colim}}\nolimits _{s : K^\bullet \to I^\bullet } \mathop{\mathrm{colim}}\nolimits _{t : P^\bullet \to M^\bullet } \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{B})}(P^\bullet , F(I^\bullet )) \\ & = \mathop{\mathrm{colim}}\nolimits _{t : P^\bullet \to M^\bullet } \mathop{\mathrm{colim}}\nolimits _{s : K^\bullet \to I^\bullet } \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{B})}(P^\bullet , F(I^\bullet )) \\ & = \mathop{\mathrm{colim}}\nolimits _{t : P^\bullet \to M^\bullet } \mathop{\mathrm{colim}}\nolimits _{s : K^\bullet \to I^\bullet } \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(G(P^\bullet ), I^\bullet ) \\ & = \mathop{\mathrm{colim}}\nolimits _{s : K^\bullet \to I^\bullet } \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(G(P^\bullet ), K^\bullet ) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(LG(M^\bullet ), K^\bullet ) \end{align*}

The first equality holds by Categories, Lemma 4.22.6. The second equality holds by the definition of morphisms in $D(\mathcal{B})$. The third equality holds by Categories, Lemma 4.14.9. The fourth equality holds because $F$ and $G$ are adjoint. The fifth equality holds by definition of morphism in $D(\mathcal{A})$. The sixth equality holds by Categories, Lemma 4.22.7. We omit the proof of functoriality. $\square$

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