Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 5.5.4.13 (Morphisms 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. Let $(C,X)$ and $(D,Y)$ be vertices of the Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Edges of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ from $(C,X)$ to $(D,Y)$ can be identified with pairs $(f,u)$, where $f: C \rightarrow D$ is an edge of the simplicial set $\operatorname{\mathcal{C}}$ and $u: \mathscr {F}(f)(X) \rightarrow Y$ is an edge of the simplicial set $\mathscr {F}(D)$ (see Example 5.4.6.12). Moreover, the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is given on edges by the construction $U(f,u) = f$.