Proposition 5.5.3.2. The simplicial set $\operatorname{\mathcal{S}}_{\ast }$ is an $\infty $-category, and the projection map $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ is a left fibration of $\infty $-categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 5.5.1.2, the simplicial set $\operatorname{\mathcal{S}}$ is an $\infty $-category. It follows that for every object $X \in \operatorname{\mathcal{S}}$, the projection map $\operatorname{\mathcal{S}}_{X/} \rightarrow \operatorname{\mathcal{S}}$ is a left fibration (Corollary 4.3.6.12). Taking $X = \Delta ^{0}$, we conclude that the projection map $\operatorname{\mathcal{S}}_{\ast } \rightarrow \operatorname{\mathcal{S}}$ is a left fibration, so that $\operatorname{\mathcal{S}}_{\ast }$ is an $\infty $-category (Remark 4.2.1.4). $\square$