Injective Cogenerator in Functor Category The existence of an injective cogenerator in $(\mathcal{A}, \mathbf{Ab})$ (where $\mathcal{A}$ is an Abelian category) is an important step in the proof of the Freyd-Mitchell Embedding Theorem and on Wikipedia it states explicitly that $\prod_{A\in \mathcal{A}}h^{A}$ is such an injective cogenerator. https://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem 
I can't really see why this is true though. It seems like the Yoneda lemma may be useful, but I can't see how to apply it. We need to check $(-,\prod_{A\in \mathcal{A}}h^{A})=\prod_{A\in \mathcal{A}}(-,h^{A})$ is an exact embedding, but the Yoneda lemma doesn't really tell us anything about $(-,h^{A})$.
Can anyone explain this to me?
 A: The category that needs to have an injective cogenerator is not $(\mathcal{A},\mathbf{Ab})$ but the full subcategory $\text{Lex}(\mathcal{A},\mathbf{Ab})$ of left exact functors.
Wikipedia claims that $\prod_{A\in\mathcal{A}}h^A$ is an injective cogenerator, but that's incorrect. In fact, Yoneda's Lemma easily shows that $\prod_{A\in\mathcal{A}}h^A$ is a generator for $\text{Lex}(\mathcal{A},\mathbf{Ab})$. That's part of showing that $\text{Lex}(\mathcal{A},\mathbf{Ab})$ is a Grothendieck category, in order to then use the fact that every Grothendieck category has an injective cogenerator.
It's easy to see that $\prod_{A\in\mathcal{A}}h^A$ is not in general injective, either in $\text{Lex}(\mathcal{A},\mathbf{Ab})$ or $(\mathcal{A},\mathbf{Ab})$. If it were, then $h^A$ would have to be injective for every $A\in\mathcal{A}$. But any non-split epimorphism $B\to A$ would induce a monomorphism $h^A\to h^B$ that would have to split (by injectivity of $h^A$). But by Yoneda's Lemma, the splitting would have to be induced by a splitting of the original epimorphism $B\to A$.
