A Grothendieck category is locally (finitely) presentable; see for example Theorem 2.2 in this paper by Positelski 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.

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.