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Corollary 1.4.3.5. Let $S_{\bullet }$ be a simplicial set having homotopy category $\mathrm{h} \mathit{S}_{\bullet }$. Then, for any category $\operatorname{\mathcal{D}}$, the composite map

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

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

Proof. The construction $S_{\bullet } \mapsto \rho _{S_{\bullet }}$ 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 the category $\operatorname{Set_{\Delta }}$ is generated under colimits by objects of the form $\Delta ^ n$ (Lemma 1.1.8.17), it will suffice to prove Corollary 1.4.3.5 in the special case where $S_{\bullet } \simeq \Delta ^ n$. In this case, the desired result follows from Proposition 1.4.3.3, since $S_{\bullet }$ is isomorphic to the nerve of the category $\operatorname{\mathcal{C}}= [n]$. $\square$