Remark 5.3.1.2. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be an inner fibration of $\infty $-categories. Then, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set
\[ \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}) \]
is an $\infty $-category (Corollary 4.1.4.8).