I think I am very confused about something.
I've been reading a bit about the Mitchell embedding theorem, and I read that the proof first embeds a given small abelian category $\mathscr{A}$ into the category of left exact functors $\mathcal{L}(\mathscr{A})$, $\mathscr{A}\to\text{Ab}$, via the Yoneda embedding. Yoneda's lemma gives that this is fully faithful, and left exactness of Hom is meant to give left exactness of the embedding (which I believe), and apparently the hard part of the proof is to show that the embedding is right exact, i.e. the Yoneda embedding $\mathscr{A}\to\mathcal{L}(\mathscr{A})$ is exact.
But if this is the case, then doesn't that mean that if $0\to a'\to a\to a''\to 0$ is exact in $\mathscr{A}$ then $0\to h_{a''}\to h_{a}\to h_{a'}\to0$ is exact (h is Yoneda embedding here) and hence $0\to$Hom$(a'',b)\to$Hom$(a,b)\to$Hom$(a',b)\to 0$ is exact for for all $b$? I guess not of course since Hom is not exact, so where am I getting tripped up? Is this not a necessary condition of exactness in $\mathcal{L}(\mathscr{A})$?
Thanks.