\[ D_{X} \colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\to \mathcal{P}\webleft (X\webright ) \]
\begin{align*} D_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [-,\text{Ø}\webright ]_{X}\\ & = \webleft (-\webright )^{\textsf{c}}\end{align*}
is an involutory isomorphism of categories, making $\text{Ø}$ into a dualising object for $\webleft (\mathcal{P}\webleft (X\webright ),\cap ,X,\webleft [-,-\webright ]_{X}\webright )$ in the sense of 
. In particular:
-
The diagram
commutes, i.e. we have
\[ \underbrace{D_{X}\webleft (D_{X}\webleft (U\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft [U,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}}=U \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
-
The diagram
commutes, i.e. we have
\[ \underbrace{D_{X}\webleft (U\cap D_{X}\webleft (V\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U\cap \webleft [V,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}}=\webleft [U,V\webright ]_{X} \]
for each $U,V\in \mathcal{P}\webleft (X\webright )$.