Definition 1.14 (Monomorphism / epimorphism). Let $f:A\to B$ be a morphism in a category $C$. We say $f$ is a monomorphism (or monic) if, given $C\underset{h}{\overset{g}{\rightrightarrows }}A$, $fg=fh⟹g=h$. We say $f$ is an epimorphism (or epic) if it’s a monomorphism in $C^{op}$.

We write $A\stackrel{f}{\rightarrowtail }B$ to indicate that $f$ is monic, and $A\stackrel{f}{\twoheadrightarrow }B$ to indicate that it’s epic.

We say $C$ is balanced if every arrow which is monic and epic is an isomorphism.