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?
