Let $$\matrix{ A& \mathop{\longrightarrow}\limits^f &B\\ \Big\downarrow & & \Big\downarrow\\ C&\mathop{\longrightarrow}\limits_g &D }$$ Be a pushout diagram in a category $\mathcal C$. If $f$ is monic, is $g$ also monic?
I have known that this holds in an abelian category. Is it true for a general category? If so, how to prove it?
If if fails, could anyone give me a counterexample? And what conditions should we impose on the category $\mathcal C$ to ensure that this is true?
Thanks!