Symmetric Lax Monoidality With Respect to Products I. The powerset functor $\mathcal{P}_{*}$ of Item 1 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
\[ \begin{gathered} \mathcal{P}^{\times }_{*|X,Y} \colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright ) \to \mathcal{P}\webleft (X\times Y\webright ),\\ \mathcal{P}^{\times }_{*|\mathbb {1}} \colon \text{pt}\to \mathcal{P}\webleft (\text{pt}\webright ), \end{gathered} \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, where
- The map $\mathcal{P}^{\times }_{*|X,Y}$ is given by
\[ \mathcal{P}^{\times }_{*|X,Y}\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U\times V \]
for each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright )$,
- The map $\mathcal{P}^{\times }_{*|\mathbb {1}}$ is given by
\[ \mathcal{P}^{\times }_{*|\mathbb {1}}\webleft (\star \webright )=\text{pt}. \]