Let A be a Grothendieck category satisfying (AB4*) (that is, with exact products).

Assuming that, moreover, A is locally noetherian, must this category have enough small projective objects (so, to be equivalent to a category of additive functors from a small preadditive category to abelian groups)?

If it is, may the "locally noetherian" assumption be relaxed into "locally finitely presented", or even "locally finitely generated" of (AB6)?

Thanks!

somecondition is needed. $\endgroup$ – Tim Campion Jun 4 at 20:14