Adjointness. We have adjunctions where
\[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (-_{1},-_{2}\webright )\colon \mathcal{P}\webleft (X\webright )\mkern -0.0mu^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \]
is the bifunctor defined by
\[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus U\webright )\cup V \]
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,\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,W\webright )\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,W\webright )\webright ), \end{align*}
natural in $U,V,W\in \mathcal{P}\webleft (X\webright )$, i.e. where:
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The following conditions are equivalent:
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We have $U\cap V\subset W$.
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We have $U\subset \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,W\webright )$.
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We have $U\subset \webleft (X\setminus V\webright )\cup W$.
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The following conditions are equivalent:
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We have $V\cap U\subset W$.
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We have $V\subset \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,W\webright )$.
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We have $V\subset \webleft (X\setminus U\webright )\cup W$.