Grothendieck categories are complete

Grothendieck categories are locally presentable, and it's a more general fact that although locally presentable categories are only required to be cocomplete, the other axioms imply that they are in fact complete.

This follows from the fact that locally presentable categories satisfy a very strong form of the adjoint functor theorem: any functor between locally presentable categories that preserves colimits has a right adjoint. Now apply this result to the diagonal functor $C \to C^J$, where $C$ is locally presentable and $J$ is a (small) diagram.

For a reference see Corollary 5.2.8 in Borceux's Handbook of Categorical Algebra Vol. II although the proof given there is different.