Category of Abelian group pairs is not Abelian
You're right that there's a simple counterexample.
Consider $f : (0, \mathbb{Z}) \to (\mathbb{Z},\mathbb{Z})$ which is the identity in the second slot. That is:
Can you show that this is a mono and an epi, but is not an iso? This will contradict the fact that every abelian category is balanced (see here, say)
I hope this helps ^_^