$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Corollary is essentially due to Eilenberg and Zilber. More precisely, in [MR52767], Eilenberg and Zilber proved that there exists a collection of quasi-isomorphisms $G_{S,T}: \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }(S \times T; \operatorname{\mathbf{Z}})$ depending functorially on the simplicial sets $S_{\bullet }$ and $T_{\bullet }$. The proof given in [MR52767] uses the method of acyclic models and does not provide a concrete description of the maps $G_{S,T}$. However, it is not difficult to see that such a collection of chain maps $\{ G_{S,T} \} $ must coincide up to sign with the Eilenberg-Zilber homomorphisms of Example (see Exercise below).