Remark 4.6.8.21. The adjunction of Corollary 4.6.8.20 has an interpretation in the framework of Proposition 1.2.3.15. Let $Q^{\bullet }$ denote the cosimplicial object of $\operatorname{Set_{\Delta }}$ given by the construction $[n] \mapsto \Phi ( \Delta ^ n )$. For every simplicial set $B$, Corollary 4.6.8.19 supplies a canonical isomorphism of simplicial sets
\[ \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}[B] )}(x,y) \simeq \operatorname{Sing}_{\bullet }^{Q}( B ), \]
where $\operatorname{Sing}_{\bullet }^{Q}(B)$ is the simplicial set defined in Variant 1.2.2.8. It follows that $\Phi $ can be identified with the generalized geometric realization functor $K \mapsto | K |^{Q}$ of Proposition 1.2.3.15.