Proposition 2.5.9.1. The functor $\mathrm{K}: \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{Set_{\Delta }}$ admits a lax monoidal structure, which associates to each pair of chain complexes $X_{\ast }$ and $Y_{\ast }$ a map of simplicial sets
\[ \mu _{X_{\ast }, Y_{\ast }}: \mathrm{K}( X_{\ast } ) \times \mathrm{K}(Y_{\ast } ) \rightarrow \mathrm{K}( X_{\ast } \boxtimes Y_{\ast } ) \]
which can be described concretely as follows:
Let $\sigma $ and $\tau $ be $n$-simplices of $\mathrm{K}(X_{\ast })$ and $\mathrm{K}(Y_{\ast })$, respectively, which we identify with chain maps
\[ \sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow X_{\ast } \quad \quad \tau : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow Y_{\ast }. \]Then $\mu _{X_{\ast }, Y_{\ast }}( \sigma , \tau ) \in \mathrm{K}_{n}( X_{\ast } \boxtimes Y_{\ast } )$ is the composite map
\[ \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \hookrightarrow \mathrm{N}_{\ast }( \Delta ^ n \times \Delta ^ n; \operatorname{\mathbf{Z}}) \xrightarrow { \mathrm{AW} } \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \xrightarrow {\sigma \boxtimes \tau } X_{\ast } \boxtimes Y_{\ast }. \]