Symmetric Strong Monoidality With Respect to Coproducts I. The powerset functor $\mathcal{P}_{*}$ of Item 1 of Proposition 2.4.3.1.4 has a symmetric strong monoidal structure
\[ \webleft (\mathcal{P}_{*},\mathcal{P}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{*},\mathcal{P}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{*|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \to \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\emptyset \webright ) \]
being equipped with isomorphisms
\[ \begin{gathered} \mathcal{P}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{*|X,Y} \colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright ) \xrightarrow {\cong }\mathcal{P}\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright ),\\ \mathcal{P}^{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{*|\mathbb {1}} \colon \text{pt}\xrightarrow {\cong }\mathcal{P}\webleft (\emptyset \webright ), \end{gathered} \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.