Kerodon

Remark 10.3.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be the sieve generated by $f$. Then $f$ is a quotient morphism if and only if the forgetful functor $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}_{/Y}$ at the object $\operatorname{id}_{Y}: Y \rightarrow Y$ (see Remark 10.3.1.32). In particular, if $f$ is a universal quotient morphism, then $f$ is a quotient morphism. Beware that the converse is false in general (see Example 10.3.4.11).