Do these two definitions of *enough projectives* coincide?

Question

Let $\mathcal{C}$ be a $k$-linear abelian category. Are the following two definitions of enough projectives equivalent?

(i) For every object $A$ there exists a projective object $P$ s.t. $P\twoheadrightarrow A$

(ii) Every simple object $U_i$ has a projective cover $p_i: P_i\twoheadrightarrow U_i$

The first one is taken from wikipedia, the second from the book Tensor Categories by Etingof et al.

I think this cannot be the same, for the superficial reason that nothing in (ii) makes a reference to arbitrary (generally indecomposable) objects of $\mathcal{C}$. However, I might be very wrong since I cannot seem to prove it.

Any pointers?

Answer 1

They do not, in general. I bet Etingof is assuming we're in a semisimple category, as is usually the case, if not always, in that book. If every object is a sum of simples, then it should be clear that these conditions are equivalent, since a sum of projectives is projective and a sum of epis is epi. The conditions are not equivalent in general, since abelian categories need not have any simple objects. See Jeremy Rickard's example here.

Answer 2

Taking into account the local finiteness assumption in your comment to Kevin Arlin's answer (which should rather be in the question itself), here is the receipe to treat a general object, assuming (ii): consider its head (quotient by the radical, which is semisimple), and take the direct sum of the projective covers of the simples involved.