Remark 5.3.1.3. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets. For each object $C \in \operatorname{\mathcal{C}}$, we can regard the identity morphism $\operatorname{id}_{C}$ as an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{C/}$. Evaluation on $\operatorname{id}_{C}$ determines a morphism of simplicial sets
Note that $\operatorname{id}_{C}$ is an initial object of the category $\operatorname{\mathcal{C}}_{C/}$, so the inclusion map $\{ \operatorname{id}_ C \} \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} )$ is left anodyne (Corollary 4.6.7.24). If $U$ is a left fibration of $\infty $-categories, then $\operatorname{ev}_{C}$ is a trivial Kan fibration of simplicial sets. It follows that the simplicial set $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is a Kan complex, and that $\operatorname{ev}_{C}$ is a homotopy equivalence of Kan complexes.