Kerodon

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Remark 8.8.1.1 (Functoriality). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet }$ denote the simplicial path category of $\operatorname{\mathcal{C}}$ (see Definition 2.4.4.1). Suppose we are given a pair of simplicial functors $\mathscr {F}, \mathscr {G}: \operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet } \rightarrow \operatorname{Set_{\Delta }}$, which (by a slight abuse of notation) we will identify with morphisms of simplicial sets from $\operatorname{\mathcal{C}}$ to the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$. Every natural transformation of simplicial functors $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ then induces a morphism of simplicial sets $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {G}$, which we will denote by $\int _{\operatorname{\mathcal{C}}} \alpha $.

Suppose that $\mathscr {F}$ and $\mathscr {G}$ take values in the simplicial subcategory $\operatorname{QCat}\subseteq \operatorname{Set_{\Delta }}$ of Construction 5.4.4.1, so that the projection maps $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ and $V: \int _{\operatorname{\mathcal{C}}} \mathscr {G} \rightarrow \operatorname{\mathcal{C}}$ are cocartesian fibrations (Proposition 5.5.4.2). It follows from Remark 5.5.4.14 that the morphism $\int _{\operatorname{\mathcal{C}}} \alpha $ carries $U$-cocartesian edges of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ to $V$-cocartesian edges of $\int _{\operatorname{\mathcal{C}}} \mathscr {G}$.