Item 2 (01K5) — 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.

Adjointness. We have adjunctions
witnessed by bijections
\begin{align*} \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,\webleft [V,W\webright ]_{X}\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,\webleft [U,W\webright ]_{X}\webright ). \end{align*}

In particular, the following statements hold for each $U,V,W\in \mathcal{P}\webleft (X\webright )$:
1. The following conditions are equivalent:
  1. We have $U\cap V\subset W$.
  2. We have $U\subset \webleft [V,W\webright ]_{X}$.
2. The following conditions are equivalent:
  1. We have $U\cap V\subset W$.
  2. We have $V\subset \webleft [U,W\webright ]_{X}$.

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