TITLE: What can we say if $A\twoheadrightarrow B$ and $A \rightarrowtail B$?
QUESTION [6 upvotes]: In some category, suppose there are two objects $A$ and $B$ such that the arrow-class $\mathsf{Hom}(A,B)$ has a monic $A \rightarrowtail B$ and an epic $A\twoheadrightarrow B$. Can we say anything interesting about $A$ and $B$?
Added: I'm also interested whether this implies anything interesting in special kinds of categories e.g abelian, closed monoidal, etc.

REPLY [2 votes]: In general you can say nothing about $A$ and $B$.
Consider, for example, in category of unital rings, the inclusion $\mathbb Z\to\mathbb Q$: it's monic and epic, but it isn't bijective.