$(A \cong B \; \; \; \wedge \;\;\; \gamma:A \rightarrowtail B) \implies \gamma:A \twoheadrightarrow B \;\;$?

If objects $A$ and $B$ (in some category $\mathcal{C}$) are isomorphic, and if some morphism $\gamma:A \rightarrow B$ is monic, does it follow that $\gamma$ is also epic?

Thanks!

-

This isn't even true in Set. For example, $f:\mathbb{N} \rightarrow \mathbb{N}$, $f(n)=n+1$ is monic but not epic.