Kerodon

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Example 5.5.6.7. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. For every object $C \in \operatorname{\mathcal{C}}$, the comparison morphism $\theta : \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ of Construction 5.5.6.1 induces a morphism of simplicial sets

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

Under the isomorphisms

\[ \mathscr {F}(C) \simeq \{ C\} \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \quad \quad \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})}( \Delta ^{0}, \mathscr {F}(C) ) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]

supplied by Examples 5.5.3.7 and 5.5.4.18, we can identify $\theta _{C}$ with the comparison map $\mathscr {F}(C) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})}( \Delta ^{0}, \mathscr {F}(C) )$ of Construction 4.6.6.3.