In Peter Freyd's book Abelian Categories he states a theorem saying that $\mathscr{L}(\mathscr{A})$ is complete and has an injective generator, with $\mathscr{L}(\mathscr{A})$ being the category of left-exact functors from a small abelian category $\mathscr{A}$ to the category of abelian groups.
My problem is that for that he states that the product of the functors of the form (A,_) for every object A in $\mathscr{A}$ is a generator for $\mathscr{L}(\mathscr{A})$, but what he shows in his book is only that the sum of those functors is a generator.
I'd be okay with this problem thinking it was just a switch up with product and sum since he states that arbitrary sums of left-exact functors are left-exact, but there lies another problem.
Freyd doesn't really prove that sums and products of left-exact functors are left-exact himself, and just uses a few lines of argument, so I took the task to try on my own way, and, well, it turns out that proving for the product seems to be very different than proving for the sum, and I could only do so for the products in the end.
So what I'd like to know is which affirmation is correct, and what Freyd meant to argue for this statement in his book. Is the product of (A,) a generator or is sum of (A,) a left-exact functor? Or is it both? I'd be pretty happy with clues as to how to prove either statement.