Corollary 1.5.3.6. The formation of homotopy categories determines a functor $\operatorname{Set_{\Delta }}\rightarrow \operatorname{Cat}$ which commutes with finite products.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since the construction $S \mapsto \mathrm{h} \mathit{S}$ preserves final objects, it will suffice to show that for any pair of simplicial sets $S$ and $T$, the canonical map $u: \mathrm{h} \mathit{( S \times T)} \rightarrow \mathrm{h} \mathit{S} \times \mathrm{h} \mathit{T}$. s an isomorphism of categories. In other words, we wish to show that for any category $\operatorname{\mathcal{C}}$, composition with $u$ induces a bijection
\[ \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S} \times \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{(S \times T)}, \operatorname{\mathcal{C}}). \]
Unwinding the definitions, we see that this map is given by the composition
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S} \times \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{S}, \operatorname{Fun}( \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{N}_{\bullet }( \operatorname{Fun}( \mathrm{h} \mathit{T}, \operatorname{\mathcal{C}}) ) ) \\ & \xrightarrow {\rho _{T} \circ } & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Fun}( T, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S \times T, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Cat}}( \mathrm{h} \mathit{ (S \times T )}, \operatorname{\mathcal{C}}), \end{eqnarray*}
where $\rho _{T}$ is the isomorphism appearing in the statement of Corollary 1.5.3.5. $\square$