Item 13 (008X) — The Clowder Project

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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`

*To be replaced with Linus Romer's Elemaints when it is released.

Symmetric Strict Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric strict monoidal structure
\[ \webleft (f_{!},f^{\otimes }_{!},f^{\otimes }_{!|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (X\webright ),\cap ,X\webright ) \to \webleft (\mathcal{P}\webleft (Y\webright ),\cap ,Y\webright ), \]

being equipped with equalities

\[ \begin{gathered} f^{\otimes }_{!|U,V} \colon f_{!}\webleft (U\cap V\webright ) \mathbin {\overset {=}{\rightarrow }}f_{!}\webleft (U\webright )\cap f_{!}\webleft (V\webright ),\\ f^{\otimes }_{!|\mathbb {1}} \colon f_{!}\webleft (X\webright ) \mathbin {\overset {=}{\rightarrow }}Y, \end{gathered} \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.

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