Kerodon

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

Remark 4.5.9.8. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets, and let $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ be the projection map of Construction 4.5.9.1. For every vertex $C \in \operatorname{\mathcal{C}}$, Remark 4.5.9.7 and Example 4.5.9.6 furnish an isomorphism of simplicial sets

\[ \pi ^{-1} \{ C\} = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{C} }(\operatorname{\mathcal{D}}_{C}, \operatorname{\mathcal{E}}_{C} ). \]