$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Example 5.6.4.7. 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 $\theta : \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ of Construction 5.6.4.1 induces a morphism of simplicial sets
\[ \theta _{C}: \{ C\} \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}). \]
Under the isomorphisms
\[ \mathscr {F}(C) \simeq \{ C\} \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \quad \quad \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Set_{\Delta }})}( \Delta ^{0}, \mathscr {F}(C) ) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \]
supplied by Examples 5.3.3.8 and 5.6.2.19, 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 4.6.8.3.