Corollary 1.5.3.5 (0064)

Corollary 1.5.3.5. Let $S$ be a simplicial set having homotopy category $\mathrm{h} \mathit{S}$. Then, for any category $\operatorname{\mathcal{D}}$, the composite map

\[ \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{S}, \operatorname{\mathcal{D}}) ) \times S \rightarrow \operatorname{N}_{\bullet }( \operatorname{Fun}(\mathrm{h} \mathit{S}, \operatorname{\mathcal{D}}) ) \times \operatorname{N}_{\bullet }(\mathrm{h} \mathit{S}) \simeq \operatorname{N}_{\bullet }( \operatorname{Fun}(\mathrm{h} \mathit{S}, \operatorname{\mathcal{D}}) \times \mathrm{h} \mathit{S}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \]

induces an isomorphism of simplicial sets $\rho _{S}: \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{S}, \operatorname{\mathcal{D}}) ) \simeq \operatorname{Fun}( S, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )$.

Proof. The construction $S \mapsto \rho _{S}$ carries colimits (in the category $\operatorname{Set_{\Delta }}$ of simplicial sets) to limits (in the category $\operatorname{Fun}([1], \operatorname{Set_{\Delta }})$ of morphisms between simplicial sets). Since every simplicial set can be realized as a colimit of standard simplices (Remark 1.1.3.13), it will suffice to prove Corollary 1.5.3.5 in the special case where $S = \Delta ^ n$ for some $n \geq 0$. In this case, the desired result follows from Proposition 1.5.3.3, since $S$ is isomorphic to the nerve of the category $\operatorname{\mathcal{C}}= [n]$. $\square$