Example 10.3.1.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. By virtue of Remark 10.3.1.14, there is a smallest sieve $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ on the object $Y$ which contains the morphism $f$. We will refer to $\operatorname{\mathcal{C}}^{0}_{/Y}$ as the sieve generated by $f$. Concretely, a morphism $e: C \rightarrow Y$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ if and only if there exists a commutative diagram
in the $\infty $-category $\operatorname{\mathcal{C}}$. Stated more informally, a morphism $e$ belongs to the sieve $\operatorname{\mathcal{C}}^{0}_{/Y}$ if and only if it factors through $f$.