Let $C$ be a category, say with finite products. What can be said about the category $Ab(C)$ of abelian group objects of $C$? Is it always an abelian category? If not, what assumptions on $C$ have to be made? What happens when $C$ is the category of smooth proper geometrically integral schemes over some locally noetherian scheme $S$?

For example if $C=Set$, we get of course the abelian category of abelian groups. If $C=Ring/R$ for some ring $R$, then we get the abelian category $Mod(R)$ (cf. nlab). In general, I have already trouble to show that $Hom(A,B) \times Hom(B,C) \to Hom(A,C)$ is linear in the left coordinate if $A,B,C$ are abelian group objects.

Let $C$ be a category, say with finite products. What can be said about the category $\operatorname{Ab}(C)$ of abelian group objects of $C$? Is it always an abelian category? If not, what assumptions on $C$ have to be made? What happens when $C$ is the category of smooth proper geometrically integral schemes over some locally noetherian scheme $S$?

For example if $C=\mathrm{Set}$, we get of course the abelian category of abelian groups. If $C=\mathrm{Ring}/R$ for some ring $R$, then we get the abelian category $\operatorname{Mod}(R)$ (cf. nLab). In general, I have already trouble to show that $\operatorname{Hom}(A,B) \times \operatorname{Hom}(B,C) \to \operatorname{Hom}(A,C)$ is linear in the left coordinate if $A$, $B$, $C$ are abelian group objects.