Kerodon

Remark 10.3.1.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on $Y$. The condition that a morphism $f: X \rightarrow Y$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$ depends only on the isomorphism class of $f$ as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. In particular, if $X$ is fixed, then the condition that $f$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$ depends only on the homotopy class $[f] \in \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X, Y)$.