# Kerodon

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Remark 5.2.3.10. 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.