Kerodon

Remark 10.3.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be the sieve generated by $X$ (Remark 10.3.1.14): that is, the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $C$ for which the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is nonempty. Then $X$ is a subterminal object of $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.3.2.2) if and only if it is a final object of $\operatorname{\mathcal{C}}^{0}$ (in the sense of Definition 4.6.7.1).