Item 6 (01J9) — The Clowder Project

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Interaction With Products III. We have an isomorphism
\[ \mathcal{P}\webleft (X\webright )\otimes \mathcal{P}\webleft (Y\webright )\cong \mathcal{P}\webleft (X\times Y\webright ), \]

natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ with respect to each of the functor structures $\mathcal{P}_{*}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{!}$ on $\mathcal{P}$ of Proposition 2.4.2.1.1, where $\otimes $ denotes the tensor product of suplattices of . Moreover, this makes $\mathcal{P}_{*}$, $\mathcal{P}^{-1}$, and $\mathcal{P}_{!}$ into symmetric monoidal functors.

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