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