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Example (Objects of the $\infty $-Category of Elements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$ be a morphism of simplicial sets. Then vertices of the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ can be identified with pairs $(C, X)$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and $X$ is a vertex of the simplicial set $\mathscr {F}(C)$ (see Example Moreover, the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is given on vertices by the construction $U(C,X) = C$.