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*}
natural in $U,V,W\in \mathcal{P}\webleft (X\webright )$, where
\[ \webleft [-_{1},-_{2}\webright ]_{X}\colon \mathcal{P}\webleft (X\webright )\mkern -0.0mu^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \]
is the bifunctor of Section 2.4.7. In particular, the following statements hold for each $U,V,W\in \mathcal{P}\webleft (X\webright )$:
-
The following conditions are equivalent:
-
We have $U\cap V\subset W$.
-
We have $U\subset \webleft [V,W\webright ]_{X}$.
-
The following conditions are equivalent:
-
We have $U\cap V\subset W$.
-
We have $V\subset \webleft [U,W\webright ]_{X}$.