I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work? Wikipedia defines the notion of an abelian category as follows (link).

A category is abelian iff
  
  
*
  
*it has a zero object,
  
*it has all binary products and binary coproducts, and
  
*it has all kernels and cokernels.
  
*all monomorphisms and epimorphisms are normal.
  

It later explains that an abelian category can naturally be enriched in $\mathbf{Ab},$ as a result of the first three axioms above.
How does this actually work?

The claim is not trivial and requires a bit of ingenuity. For a guided solution, see Q6 here. The four steps are as follows:


*

*Show that the category has finite limits and finite colimits.

*Show that a morphism $f$ is monic (resp. epic) if and only if $\ker f = 0$ (resp. $\operatorname{coker} f = 0$).

*Show that the canonical morphisms $A + B \to A \times B$ are isomorphisms, and then deduce that the category is enriched over commutative monoids.

*Show that every morphism has an additive inverse (by inverting an appropriate matrix).


A complete proof is given as Theorem 1.6.4 in [Borceux, Handbook of categorical algebra, Vol. 2].
