Exercise 2.5.7.18. For every pair of simplicial sets $S_{\bullet }$ and $T_{\bullet }$, let
\[ G_{S,T}: \mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}( S \times T; \operatorname{\mathbf{Z}}) \]
be a chain map. Assume that the maps $G_{S,T}$ depend functorially on $S_{\bullet }$ and $T_{\bullet }$: that is, for all maps of simplicial sets $f: S_{\bullet } \rightarrow S'_{\bullet }$ and $g: T_{\bullet } \rightarrow T'_{\bullet }$, the diagram of chain complexes
\[ \xymatrix@R =50pt@C=50pt{ \mathrm{N}_{\ast }( S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T; \operatorname{\mathbf{Z}}) \ar [r]^-{ G_{S,T}} \ar [d]^{ \mathrm{N}_{\ast }(f; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(g; \operatorname{\mathbf{Z}}) } & \mathrm{N}_{\ast }(S \times T; \operatorname{\mathbf{Z}}) \ar [d]^{ \mathrm{N}_{\ast }(f \times g; \operatorname{\mathbf{Z}}) } \\ \mathrm{N}_{\ast }( S'; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T'; \operatorname{\mathbf{Z}}) \ar [r]^-{ G_{S',T'}} & \mathrm{N}_{\ast }(S' \times T'; \operatorname{\mathbf{Z}}) } \]
is commutative. Adapt the proof Proposition 2.5.7.1 to show that there exists an integer $n$ (not depending on $S_{\bullet }$ and $T_{\bullet }$) such that $G_{S,T} = n \mathrm{EZ}$, where $\mathrm{EZ}$ is the Eilenberg-Zilber homomorphism of Example 2.5.7.12.