Symmetric Strong Monoidality With Respect to Smash Products. The free pointed set functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )^{-},\webleft (-\webright )^{-,\times },\webleft (-\webright )^{-,\times }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets}^{\mathrm{actv}}_{*},\wedge ,S^{0}\webright ), \to \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \]
being equipped with isomorphisms of pointed sets
\[ \begin{gathered} \webleft (-\webright )^{-,\times }_{X,Y} \colon X^{-}\times Y^{-} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\wedge Y\webright )^{-},\\ \webleft (-\webright )^{-,\times }_{\mathbb {1}} \colon \text{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (S^{0}\webright )^{-}, \end{gathered} \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.