Corollary 4.7.3.13. Let $X$ be a Kan complex and let $x \in X$ be a vertex. The following conditions are equivalent:
The vertex $x$ is initial when viewed as an object of the $\infty $-category $X$.
The vertex $x$ is final when viewed as an object of the $\infty $-category $X$.
The Kan complex $X$ is contractible.
In particular, these conditions are independent of the choice of vertex $x \in X$.
Proof. If the Kan complex $X$ is contractible, then the projection map $X_{/x} \rightarrow X$ is a trivial Kan fibration (Corollary 4.3.7.19), so the object $x \in X$ is final by virtue of Corollary 4.7.3.12. Conversely, if the projection map $X_{/x} \rightarrow X$ is a trivial Kan fibration, then it is a homotopy equivalence (Proposition 3.1.7.10). Since the Kan complex $X_{/x}$ is contractible (Corollary 4.3.7.14), it follows that $X$ is contractible. This proves the equivalence of $(2) \Leftrightarrow (3)$; the equivalence of $(1) \Leftrightarrow (3)$ follows by a similar argument. $\square$