Elements and arrows in an abelian category. Suppose to work in an abelian category $\mathcal{A}$, so in particular for every objects $A$ and $B$, we have that $Hom(A,B)$ is an abelian group - in particular a set. My questions are: 1. Does it suffice to conclude that $A$ and $B$ are sets? 2. If "yes", does it suffice to conclude that the arrows from $A$ to $B$ are function, in a set-theoretic sense? I am almost sure that both of the statement are _completely wrong_ , but I should be better if I received a proof/disproof (if possible, an example should be perfect!). Thanks in advance. Cheers

The formal answer is no, the objects of an abelian category need not be sets, nor do the morphisms need be functions. However, the Freyd-Mitechell embedding theorem says that an abelian category, and particularly a small one, is not far from being essentially a category of modules over some ring, with morphisms the module homomorphisms. In particular, when dealing with small enough diagrams in an abelian category one can safely pretend one is working with modules.