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Example 5.5.4.15 ($2$-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

\[ \xymatrix@R =50pt@C=50pt{ & (D,Y) \ar [dr]^{ (g,v) } & \\ (C,X) \ar [ur]^{(f,u)} \ar [rr]^{ (h,w) } & & (E,Z). } \]

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

\[ \xymatrix@R =50pt@C=50pt{ (\mathscr {F}(g) \circ \mathscr {F}(f))(X) \ar [r]^-{\mu (X)} \ar [d]^-{ \mathscr {F}(g)(u)} & \mathscr {F}(h)(X) \ar [d]^-{w} \\ \mathscr {F}(g)(Y) \ar [r]^-{v} & Z } \]

(see Example 5.4.6.17). Moreover, the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \mathscr {C}$ is given on $2$-simplices by the construction $U( \mu , \theta ) = \mu $.