Warning 5.6.4.4. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. The comparison map $\theta : \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ of Construction 5.6.4.1 is generally not bijective on $n$-simplices for $n \geq 2$ (even in the special case $\operatorname{\mathcal{C}}= [0]$).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$