Corollary 4.7.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then:
- $(1)$
The object $X$ is initial if and only if the projection map $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets.
- $(2)$
The object $X$ is final if and only if the projection map $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration of simplicial sets.