• 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 $U\in \mathcal{P}\webleft (X\webright )$.


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