Item 2 (00RV) — The Clowder Project

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Adjointness. We have an adjunction
witnessed by a bijections of sets
\[ \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (R_{*}\webleft (U\webright ),V\webright )\cong \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (U,R_{-1}\webleft (V\webright )\webright ), \]

natural in $U\in \mathcal{P}\webleft (A\webright )$ and $V\in \mathcal{P}\webleft (B\webright )$, i.e. such that:
- The following conditions are equivalent:
  - We have $R_{*}\webleft (U\webright )\subset V$.
  - We have $U\subset R_{-1}\webleft (V\webright )$.

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