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Corollary 1.4.3.6. The formation of homotopy categories determines a functor $\operatorname{Set_{\Delta }}\rightarrow \operatorname{Cat}$ which commutes with finite products.

Proof. Since the construction $S_{\bullet } \mapsto \mathrm{h} \mathit{S}_{\bullet }$ preserves final objects, it will suffice to show that for any pair of simplicial sets $S_{\bullet }$ and $T_{\bullet }$, the canonical map

\[ u: \mathrm{h} \mathit{( S_{\bullet } \times T_{\bullet } )} \rightarrow \mathrm{h} \mathit{S}_{\bullet } \times \mathrm{h} \mathit{T}_{\bullet } \]

is an isomorphism of categories. In other words, we wish to show that for any category $\operatorname{\mathcal{C}}$, composition with $u$ induces a bijection

\[ \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}_{\bullet } \times \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{(S_{\bullet } \times T_{\bullet })}, \operatorname{\mathcal{C}}). \]

Unwinding the definitions, we see that this map is given by the composition

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}_{\bullet } \times \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}_{\bullet }, \operatorname{Fun}( \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{T}_{\bullet }, \operatorname{\mathcal{C}}) ) ) \\ & \xrightarrow {\rho _{T_{\bullet }} \circ } & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Fun}( T_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet } \times T_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{ (S_{\bullet } \times T_{\bullet } )}, \operatorname{\mathcal{C}}), \end{eqnarray*}

where $\rho _{T_{\bullet }}$ is the isomorphism appearing in the statement of Corollary 1.4.3.5. $\square$