Example 5.6.2.15 ($2$-Simplices 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 and let $\sigma _0: \operatorname{\partial \Delta }^2 \rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be a morphism of simplicial sets, which we depict informally as a diagram
Extensions of $\sigma _0$ to a $2$-simplex of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ can be identified with pairs $(\mu , \theta )$, where $\mu : \mathscr {F}(g) \circ \mathscr {F}(f) \rightarrow \mathscr {F}(h)$ is an edge of the simplicial set $\operatorname{Fun}( \mathscr {F}(C), \mathscr {F}(E) )$, and $\theta : \operatorname{\raise {0.1ex}{\square }}^{2} \rightarrow \mathscr {F}(E)$ is a morphism of simplicial sets whose restriction to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is indicated in the diagram
(see Example 5.5.6.17). Moreover, the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is given on $2$-simplices by the construction $U( \mu , \theta ) = \mu $.