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?