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