Variant 6.3.7.4. Let $S$ be a simplicial set and let $\psi _{S}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow \operatorname{Sd}(S)$ be the comparison map of Construction 3.3.3.10. Then $\psi _{S}$ is universally localizing.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Note that the functor $S \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} )$ preserves small colimits (Variant 3.3.3.20). Proceeding as in the proof of Proposition 6.3.7.2, we can reduce to the case where $S = \Delta ^{n}$ is a standard simplex. In this case, we can identify $\psi _{S}$ with (the nerve of) the functor
\[ \operatorname{{\bf \Delta }}_{S} \rightarrow \operatorname{Chain}[n] \quad \quad (\alpha : [m] \rightarrow [n] \mapsto \mathrm{im}(\alpha ) \subseteq [n] ) \]
This functor admits a section $\phi $, which identifies $\operatorname{Chain}[n]$ with the subcategory $\operatorname{{\bf \Delta }}_{S}^{\mathrm{nd}} \subseteq \operatorname{{\bf \Delta }}_{S}$ of nondegenerate simplices of $S$. Note that there is a (unique) natural transformation from the identity functor $\operatorname{id}_{ \operatorname{{\bf \Delta }}_{S} }$ to $\phi \circ \psi _{S}$ which belongs to $\operatorname{Fun}_{ / \operatorname{Sd}(S) }( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ), \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) )$, so the desired result follows from the criterion of Proposition 6.3.6.8. $\square$