Example 10.3.0.4. Let $\operatorname{\mathcal{C}}= \operatorname{Set}$ be the category of sets and let $f: X \twoheadrightarrow Y$ be an epimorphism in $\operatorname{\mathcal{C}}$: that is, a surjective function. Then $f$ is a regular epimorphism: that is, it exhibits $Y$ as a quotient of the equivalence relation $\equiv _{f}$, defined by the requirement
\[ (x \equiv _{f} x' ) \Leftrightarrow ( f(x) = f(x') ). \]