The smash product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$1 is the pointed set $X\wedge Y$2 satisfying the bijection
\[ \mathsf{Sets}_{*}\webleft (X\wedge Y,Z\webright ) \cong \textup{Hom}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ), \]
naturally in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
1Further Terminology: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, the smash product $X\wedge Y$ is also called the tensor product of $\mathbb {F}_{1}$-modules of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ or the tensor product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ over $\mathbb {F}_{1}$.
2Further Notation: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, the smash product $X\wedge Y$ is also denoted $X\otimes _{\mathbb {F}_{1}}Y$.
