I expect this to be a small understanding problem and not a real interesting question.

In Gabriel's thesis you find a proof of the theorem that every small abelian category $C$ admits a faithful exact functor to the category of abelian groups. The proof goes like this: Consider the abelian category $Sex(C,Ab)$, which is the category of left-exact functors $C \to Ab$. By the way, does anybody know why Gabriel has chosen this notation? Why not "Lex" for left-exact or "Gex" for the french "exacte a gauche"? Anyway, it can be shown that $Sex(C,Ab)$ is a (nowadays called) Grothendieck category and has thus enough injective objects. Besides, every injective object is an exact functor $C \to Ab$. If we embed the generator $U=\sum_{X \in C} Hom(X,-)$ into an injective object, we are done.

Now Gabriels claims that $U$ is actually a projective generator. I doubt that this is used in the proof, nevertheless it is interesting. By Yoneda the functor $F \mapsto Hom(U,F)$ is isomorphic to $F \mapsto \prod_{X \in C} F(X)$, thus we have to prove that every epimorphism in $Sex(C,Ab)$ is actually pointwise an epimorphism of abelian groups. This is not clear to me since a cokernel in $Sex(C,Ab)$ is defined by the universal left-exact functor associated to the pointwise-defined cokernel (cf. Prop. 5 in II.2). The vanishing does not seem to imply the vanishing of the pointwise-defined cokernel (cf. Lemme 3 b).

Note that Grothendieck has supervised this thesis. Although there are many many typos, I don't think that such a statement will just be wrong. What am I missing?