# Can monomorphisms between two objects imply isomorphism?

I have a very basic understanding regarding category theory, but I was wondering whether there is a condition for isomorphism between objects via monomorphisms? More concretely, given a category $$\mathcal{C}$$ and two objects $$A$$ and $$B$$, such that we have monomorphisms $$A\overset{f}{\rightarrow}B$$ and $$B\overset{g}{\rightarrow}A$$, are there conditions implying the existence of an isomorphism $$A\overset{h}{\rightarrow}B$$?

By what I understand from searching online, if $$\mathcal{C}$$ is a balanced category this should be true given that monomorphisms are injective functions. Is there some stronger conditions, for us to be able to say something of the sort in non balanced categories?

• It can actually fail even for balanced categories ; it fails in groups for example. – Arnaud D. Feb 5 '20 at 12:13
• This Math.SE question and this MO question are related. – Arnaud D. Feb 5 '20 at 12:15
• So my question would be more appropriately phrased as conditions for "Schroeder-Bernstein" property for non Balanced categories? – Keen-ameteur Feb 5 '20 at 12:19