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Let $F:\mathcal C\longrightarrow \mathcal D$ be an additive functor that preserves colimits.

Suppose that $\mathcal C$ and $\mathcal D$ are Grothendieck categories.

Does $F$ have a right adjoint?

I know the adjoint functor theorems. But when I checked the exact statements in standard references like Kashiwara Schapira, it is not clear to me whether the results apply to additive functors.

Could someone help clarify with references?

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    $\begingroup$ Notice that, in Abelian categories, finite products and finite coproducts are the same; furthermore, a functor is additive if and only if it preserves finite coproducts. Hence, any functor between two Grothendieck categories that commutes either with limits or with colimits is necessarily additive. Hence, for a functor between Grothendieck categories to have any hope to be a (left or right) adjoint, it has to be additive. As a consequence: additivity here is not an hypothesis you have to add, it follows naturally by the rest of hypotheses. $\endgroup$
    – Simone Virili
    Commented Aug 6, 2019 at 17:04

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A Grothendieck category is locally (finitely) presentable; see for example Theorem 2.2 in this paper by Positselski and Rosický. And any colimit-preserving functor between locally presentable categories has a right adjoint. One way to see this is by citing Theorem 1.58 in Locally Presentable and Accessible Categories which shows that locally presentable categories are co-well-powered, and then applying the Special Adjoint Functor Theorem. Another way is to observe that locally presentable categories are total categories, and applying the adjoint functor theorem for total categories; see the nLab.

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