Kerodon

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

Example 5.5.3.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. For every morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, Remark 5.5.3.5 and Example 5.5.3.3 supply an isomorphism of simplicial sets

\[ \mathscr {F}(C) \star _{ \mathscr {F}(D) } \mathscr {F}(D) \simeq \Delta ^1 \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}. \]