Kerodon

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

Remark 5.2.4.10. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a fixed morphism of simplicial sets. Then the construction

\[ (F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}) \mapsto \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]

carries colimits in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{E}}}$ to colimits in the category $(\operatorname{Set_{\Delta }})_{\operatorname{\mathcal{D}}/}$. In particular, the construction $\operatorname{\mathcal{C}}\mapsto (\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$ commutes with filtered colimits and carries pushout diagrams to pushout diagrams.