It is well known that for a ring $R$ any projective $R$-module is the summand of a free $R$-module.
Let now $(X,\mathcal{O})$ be a site with a sheaf of rings. Are projective $\mathcal{O}$-premodules and $\mathcal{O}$-modules also summands of some $\mathcal{O}^I$ for some set $I$?
Usually the proof in the affine case starts by picking generators of the projective module $P$. So I guess the question comes down to: Does every $\mathcal{O}$-premodule $A$ admit an epimorphism $\mathcal{O}^I\to A$? And what do we do in the case of sheaves, where there are not even enough projectives?