Symmetric Strong Monoidality With Respect to Wedge Sums. The free pointed set functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )^{+},\webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}},\webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset \webright ) \to \webleft (\mathsf{Sets}_{*},\vee ,\text{pt}\webright ), \]
being equipped with isomorphisms of pointed sets
\[ \begin{gathered} \webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y} \colon X^{+}\vee Y^{+} \xrightarrow {\cong }\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )^{+},\\ \webleft (-\webright )^{+,\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{\mathbb {1}} \colon \text{pt}\xrightarrow {\cong }\emptyset ^{+}, \end{gathered} \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.