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?