Item 1 (00RC) — 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.

Functoriality. The assignment $U\mapsto R_{*}\webleft (U\webright )$ defines a functor
\[ R_{*}\colon \webleft (\mathcal{P}\webleft (A\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (B\webright ),\subset \webright ) \]

where
- Action on Objects. For each $U\in \mathcal{P}\webleft (A\webright )$, we have
  
  \[ \webleft [R_{*}\webright ]\webleft (U\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{*}\webleft (U\webright ); \]
- Action on Morphisms. For each $U,V\in \mathcal{P}\webleft (A\webright )$:
  - If $U\subset V$, then $R_{*}\webleft (U\webright )\subset R_{*}\webleft (V\webright )$.

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