Complete but not cocomplete category

I know that a consequence of the Gabriel-Popescu theorem (i.e., every Grothendieck category is a torsion-theoretic localization of a full category of modules) is that any Grothendieck category (which by definition is cocomplete) is complete. I guess that this is not true for general abelian categories, so here is the question:

is it true that a cocomplete abelian category is complete? If no (as I suppose) is there some canonical counterexample?

Solution 1:

I think it's a good idea to take questions off the unanswered list whenever possible. Here is a direct link to the MO answer: https://mathoverflow.net/a/184486/2926.