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Remark The adjunction of Corollary has an interpretation in the framework of Proposition 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 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 It follows that $\Phi $ can be identified with the generalized geometric realization functor $K \mapsto | K |^{Q}$ of Proposition