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](http://ncatlab.org/nlab/show/module#DefWithOCat)). 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.



The category of abelian group objects
Is $\mathscr{C}$ is regular/(exact in Barr sense) then for any algeraic theory $T$ the category $T{\operatorname{-Alg}}(\mathscr{C})$ of internal $T$-algebras is regular/(exact), in particular for $\mathscr{C}$ exact and $\operatorname{Ab} ={}$"commutative groups theory" we have $\operatorname{Ab}(\mathscr{C})$ is exact, this is also additive (i.e. abelian, a category is abelian iff is additive and exact):


given $f, g: A\to B$ in $\operatorname{Ab}(\mathscr{C})$ get $f+g$ in either of the following two ways:


1. appling the Yoneda valuation $h^X$, $X\in \mathscr{C}$ to $f, g: A\to B$ (and considering Yoneda's Lemma).
2. $f+g: A \xrightarrow{(f, g)} B\times B \xrightarrow{+}B$


