Remark 10.3.1.10. 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 $X$. Then $U$ restricts to a functor $U^{0}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}^{0}$. Moreover, the following conditions are equivalent:
- $(1)$
The object $X \in \operatorname{\mathcal{C}}$ is subterminal (see Definition 9.3.2.2).
- $(2)$
The functor $U^{0}$ is a trivial Kan fibration.
- $(3)$
The functor $U^{0}$ is an equivalence of $\infty $-categories.
The equivalence $(1) \Leftrightarrow (2)$ is a reformulation of Remark 9.3.2.14. Note that $U^0$ is a pullback of $U$, and therefore a right fibration (Proposition 4.3.6.1). In particular, $U^{0}$ is an isofibration (Example 4.4.1.11), so the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 4.5.5.20.