Kerodon

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

Proposition 4.5.9.2. Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every morphism of simplicial sets $\sigma : \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, postcomposition with the evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$ of Construction 4.5.9.1 induces a bijection

\[ \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}} }( \operatorname{\mathcal{C}}', \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Hom}_{ (\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{D}}} }( \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}). \]

Proof. Writing $\operatorname{\mathcal{C}}'$ as a colimit of simplices, we may reduce to the case where $\operatorname{\mathcal{C}}' = \Delta ^{n}$, so that $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$. In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$. $\square$