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Construction 5.5.3.16. Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, and let $\operatorname{N}_{\bullet }^{+}( \operatorname{Set_{\Delta }})$ be the simplicial set introduced Notation 5.2.6.1. Unwinding the definitions, we see that $n$-simplices of the fiber product $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }})$ can be identified with pairs $( \overrightarrow {C}, \sigma )$, where $\overrightarrow {C} = (C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n)$ is an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and $\sigma $ is an $n$-simplex of the simplicial set $\mathscr {F}(C_0)$. To every such pair $(\overrightarrow {C}, \sigma )$, we can associate an $n$-simplex $( \overrightarrow {C}, \overrightarrow {\sigma } )$ of $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$, where $\overrightarrow {\sigma }$ assigns to each $0 \leq i \leq n$ the $i$-simplex $\sigma _{i}: \Delta ^{i} \rightarrow \mathscr {F}(C_ i)$ given by the composition

\[ \Delta ^{i} \hookrightarrow \Delta ^{n} \xrightarrow {\sigma } \mathscr {F}(C_0) \rightarrow \mathscr {F}(C_ i). \]

The construction $( \overrightarrow {C}, \sigma ) \rightarrow (\overrightarrow {C}, \overrightarrow {\sigma } )$ determines a morphism of simplicial sets

\[ F: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})} \operatorname{N}_{\bullet }^{+}(\operatorname{Set_{\Delta }}) \rightarrow \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}. \]