Kerodon

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Construction 8.8.1.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet }$ denote the simplicial path category of $\operatorname{\mathcal{C}}$ (Notation 2.4.4.2) and let $\mathscr {F}: \operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet } \rightarrow \operatorname{Set_{\Delta }}$ be a simplicial functor. For every simplicial set $X$, we let $\mathscr {F}^{X}: \operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet } \rightarrow \operatorname{Set_{\Delta }}$ denote the simplicial functor given on objects by the formula $\mathscr {F}^{X}(C) = \operatorname{Fun}(X, \mathscr {F}(C) )$. Note that evaluation determines a natural transformation of simplicial functors $\operatorname{ev}: \underline{X} \times \mathscr {F}^{X} \rightarrow \mathscr {F}$, where $\underline{X}: \operatorname{Path}[\operatorname{\mathcal{C}}]_{\bullet } \rightarrow \operatorname{Set_{\Delta }}$ is the constant (simplicial) functor taking the value $X$.

Let us abuse notation by identifying $\mathscr {F}$, $\mathscr {F}^{X}$, and $\underline{X}$ with morphisms from $\operatorname{\mathcal{C}}$ to the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})$. Let $\theta _ X: X \rightarrow \int _{\Delta ^0} X$ be the comparison map of Example 5.5.4.16. Applying Remark 8.8.1.1, we obtain a morphism of simplicial sets

\begin{eqnarray*} X \times \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{X} & \xrightarrow {\theta _ X \times \operatorname{id}} & (\int _{\Delta ^{0}} X) \times (\int _{\operatorname{\mathcal{C}}} \mathscr {F}^{X} ) \\ & \simeq & (\operatorname{\mathcal{C}}\times \int _{\Delta ^{0}} X) \times _{\operatorname{\mathcal{C}}} (\int _{\operatorname{\mathcal{C}}} \mathscr {F}^{X} ) \\ & \simeq & ( \int _{\operatorname{\mathcal{C}}} \underline{X} ) \times _{\operatorname{\mathcal{C}}} ( \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{X} ) \\ & \simeq & \int _{\operatorname{\mathcal{C}}} ( \underline{X} \times \mathscr {F}^{X} ) \\ & \xrightarrow {\int _{\operatorname{\mathcal{C}}} \operatorname{ev}} & \int _{\operatorname{\mathcal{C}}} \mathscr {F} \end{eqnarray*}

which we can identify with a comparison map

$\theta : \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{X} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(X, \operatorname{\mathcal{C}}) } \operatorname{Fun}( X, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$

in the category $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$.