The Stacks project

Proposition 19.15.2. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $\mathcal{D}$ be a triangulated category. Let $F : D(\mathcal{A}) \to \mathcal{D}$ be an exact functor of triangulated categories which transforms direct sums into direct sums. Then $F$ has an exact right adjoint.

Proof. For an object $Y$ of $\mathcal{D}$ consider the contravariant functor

\[ D(\mathcal{A}) \to \textit{Ab},\quad W \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(F(W), Y) \]

This is a cohomological functor as $F$ is exact and transforms direct sums into products as $F$ transforms direct sums into direct sums. Thus by Lemma 19.15.1 we find an object $X$ of $D(\mathcal{A})$ such that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(W, X) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(F(W), Y)$. The existence of the adjoint follows from Categories, Lemma 4.24.2. Exactness follows from Derived Categories, Lemma 13.7.1. $\square$

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