Symmetric Lax Monoidality With Respect to Products III. The powerset functor $\mathcal{P}_{!}$ of Item 3 of Proposition 2.4.3.1.4 has a symmetric lax monoidal structure
\[ \webleft (\mathcal{P}_{!},\mathcal{P}^{\otimes }_{!},\mathcal{P}^{\otimes }_{!|\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \to \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \]
being equipped with morphisms
\[ \begin{gathered} \mathcal{P}^{\times }_{!|X,Y} \colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright ) \to \mathcal{P}\webleft (X\times Y\webright ),\\ \mathcal{P}^{\times }_{!|\mathbb {1}} \colon \text{pt}\to \mathcal{P}\webleft (\emptyset \webright ), \end{gathered} \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, defined as in Item 4.