Symmetric Lax Monoidality With Respect to Products II. The powerset functor $\mathcal{P}^{-1}$ of Item 2 of Proposition 2.4.3.1.4 has a symmetric lax monoidal structure
\[ \webleft (\mathcal{P}^{-1},\mathcal{P}^{-1|\otimes },\mathcal{P}^{-1|\otimes }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets}^{\mathsf{op}},\times ^{\mathsf{op}},\text{pt}\webright ) \to \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \]
being equipped with morphisms
\[ \begin{gathered} \mathcal{P}^{-1|\times }_{X,Y} \colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright ) \to \mathcal{P}\webleft (X\times Y\webright ),\\ \mathcal{P}^{\times }_{\mathbb {1}} \colon \text{pt}\to \mathcal{P}\webleft (\emptyset \webright ), \end{gathered} \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, defined as in Item 4.