Corollary 10.3.2.7. Let $\operatorname{\mathcal{C}}$ be a category which admits pullbacks and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $f$ is a quotient morphism in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.3.2.1).
- $(2)$
The morphism $f$ is a regular epimorphism: that is, it exhibits $Y$ as a coequalizer of the projection maps $X \times _{Y} X \rightrightarrows X$ (Definition 10.3.0.2).