I have a few elementary questions about the Freyd-Mitchell embedding theorem which I can't see answered elsewhere here and which all arise from some confusions I ran into.

Can someone point out why (and where in the proof) exactly you need that the abelian category $\mathcal{A}$ is

*small*? Is it to show that the functor category $\mathsf{Fun}(\mathcal{A},\mathsf{Ab})$ is Grothendieck?Does the FH embedding theorem also hold if $\mathcal{A}$ is only

*essentially*small? I would say so but have seen this nowhere written out. If $\mathcal{A}$ is essentially small, there is an equivalence $F \colon \mathcal{A} \to \mathcal{A}'$ into a small category $\mathcal{A}'$. The category $\mathcal{A}'$ is automatically abelian and $F$ is an exact equivalence. Composed with the FH embedding of $\mathcal{A}'$ we get an exact embedding of $\mathcal{A}$ into a module category. Where's the mistake?This one really confuses me. I thought that an exact embedding $F \colon \mathcal{A} \to R\text{-}\mathsf{Mod}$ induces an isomorphism $\mathrm{Sub}(X) \to \mathrm{Sub}(F(X))$ between the lattices of subobjects of $X$ and $F(X)$, therefore maps simple objects to simple modules etc., and I can deduce the Jordan–Hölder theorem for a finite length object in $\mathcal{A}$ simply from the one for modules. But the comments here seem to indicate that this is wrong. Why is it so? I must be stupid here.