TITLE: Does every functor between Grothendieck categories have adjoints?
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: 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.