The (internal version of) axiom of choice, stated in the language of category theory says that every epimorphism splits - i.e., every epimorphism has a section.
Now, just like any other category theoretic statement, it has a dual statement which can be produced by inverting all of the arrows. Thus, dual of axiom of choice, we which we dub axiom of co-choice in this question, states that every monomorphism is split -- i.e. has a retraction.
Now, without any assumptions beyond ZF, this holds in the topos Set. However, like axiom of choice, it fails in Top. Off the top of my head, I don't know it failing in any other relevant category.
However, despite looking for some more information on it, there seems to be no literature studying axiom of co-choice. So, my question are these:
Is the axiom of co-choice not mathematically interesting? Is there any philosophical or mathematical justification for the dual of axiom of choice - a very interesting problem on its own - being boring? If it is truly a boring or trivial result, what classes of categories do satisfy it - i.e. what are some statements or requirements which are equivalent to it?