Proposition 13.39.2. Let $\mathcal{D}$ be a triangulated category with direct sums. Assume there exists a set $\mathcal{E}$ of objects of $\mathcal{D}$ satisfying conditions (1) and (2) of Lemma 13.39.1. Let $F : \mathcal{D} \to \mathcal{D}'$ 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

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

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 13.39.1 we find an object $X$ of $\mathcal{D}$ such that $\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, X) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}'}(F(W), Y)$. The existence of the adjoint follows from Categories, Lemma 4.24.2. Exactness follows from Lemma 13.7.1. $\square$

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