Grothendieck categories are complete I've been reading a book on triangulated categories and derived categories. The writer supposes that the readers are familiar with Grothendieck categories. He lists several basic properties. The one disturbs me is that a Grothendieck category is complete. The writer refers to the book Abelian Categories Murfet
for the proof. Murfet writes before Theorem 22:

Proofs of the following results can be found in any decent reference
on category theory.

But I failed to find such one.
 A: 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. 
A: It's not necessary to invoke Gabriel-Popescu embedding theorem here.
The fact essentially follows from Special Adjoint Functor Theorem.

Special Adjoint Functor Theorem (SAFT). Let $F\colon\mathsf{C}\to\mathsf{D}$ be a continuous functor with $\mathsf{C}$ being complete well-powered category with a cogenerating set. Then $F$ admits a left adjoint.

The proof can be found in many places, such as Mac Lane or Borceux.
From this the following proposition can be deduced:

Let $\mathsf{C}$ be a complete well-powered category with a cogenerating set. Then $\mathsf{C}$ is a cocomplete.

The proof uses the fact that a category $\mathsf{C}$ admits all $\mathsf{I}$-shaped colimits if and only if the diagonal functor $\Delta\colon \mathsf{C}\to [\mathsf{I},\mathsf{C}]$ (which maps an object to its constant functor and a morphism to its constant natural transformation) has a left adjoint (which would be the colimit functor).
Then the claim that Grothendieck categories are complete follows from the dual of the above proposition (for Grothendieck categories being co-well-powered, there's an easy theorem that abelian categories are well-powered if and only of they are co-well-powered).
