I'm beginning to study abelian categories. The definition I'm using is this:
A category is said to be abelian if:
$1)$ it has a zero object
$2)$ it has all binary products and binary coproducts
$3)$ it has all kernels and cokernels
$4)$ all monomorphisms and epimorphisms are normal.
I'm trying to prove that if a category $C$ is abelian, than for every $A,B\in\text{Ob}(C)$, the set $\hom_C(A,B)$ has an abelian group structure with neutral element $0:A\to B$ (the zero morphism), and that the composition of morphism is bilinear with respect to this group operation.
In simple examples like the category of $R$-modules or $K$-vector spaces, there is an obvious additive group structure, but in the general context, I have no idea how to define this group operation.
What is the idea?