Item 5 (0088) — 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 Unions. The inverse image function of Item 1 has a symmetric strict monoidal structure
\[ \webleft (f^{-1},f^{-1,\otimes },f^{-1,\otimes }_{\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (B\webright ),\cup ,\emptyset \webright ) \to \webleft (\mathcal{P}\webleft (A\webright ),\cup ,\emptyset \webright ), \]

being equipped with equalities

\[ \begin{gathered} f^{-1,\otimes }_{U,V} \colon f^{-1}\webleft (U\webright )\cup f^{-1}\webleft (V\webright ) \mathbin {\overset {=}{\rightarrow }}f^{-1}\webleft (U\cup V\webright ),\\ f^{-1,\otimes }_{\mathbb {1}} \colon \emptyset \mathbin {\overset {=}{\rightarrow }}f^{-1}\webleft (\emptyset \webright ), \end{gathered} \]

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

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