Given a small preadditive category \mathcal{C}, the functor category \operatorname{Add}(\mathcal{C},\mathrm{Ab}) of all additive covariant (or alternatively: contravariant) functors from \mathcal{C} to the category \mathrm{Ab} of all abelian groups is a Grothendieck category.
Let \mathcal{C} be a small abelian category.