Kerodon

Example 7.4.2.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet }$ denote the simplicial path category of $\operatorname{\mathcal{C}}$ (Notation 2.4.4.2). In this case, we can identify diagrams $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ with simplicially enriched functors $\mathscr {F}, \mathscr {F}': \operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet } \rightarrow \operatorname{Kan}$. Suppose that $\gamma : \mathscr {F} \rightarrow \mathscr {F}'$ arises from a natural transformation of simplicial functors $\gamma _0: \mathscr {F}_0 \rightarrow \mathscr {F}'_0$. Applying Construction to $\gamma _0$, we obtain a map $\Gamma : \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}'$ which is compatible with $\gamma $.