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!