A category is said to be balanced in case that every bimorphism is an isomorphism.

For example, Set (the cateogry of sets), Grp (the category of groups), and Top (the category of topological spaces) are all balanced. In these categories, morphisms that are monomorphic are injective, and those that are epimorphic are surjective. Injective and surjective morphisms are bijective, implying having two-sided inverses, and therefore isomorphisms.

On the other hand, the category DivAbGrp, the category of divisible abelian groups, is not balanced. The canonical projection $p:\mathbb{Q}\to \mathbb{Q/Z}$ is clearly epimorphic, as well as monomorphic (see here), and therefore bimorphic. However, $p$ is not an isomorphism.

As another example of a category that is not balanced, take the category of commutative rings with 1, CommRng. The canonical injection $i:\mathbb{Z}\to \mathbb{Q}$ is clearly monomorphic, as well as epimorphic (see here), and therefore bimorphic. But $i$ is not an isomorphism.

Bibliography

1

C. Faith Algebra: Rings, Modules, and Categories I, Springer-Verlag, New York (1973)