Connecting the profinite completion and the canonical extension using duality
Jacob Vosmaer

Abstract:
We show using duality and category theory that the profinite
  completion $\mathbb{\hat A}$ of a bounded distributive lattice
  expansion $\mathbb{A}$ is a homomorphic image of the canonical
  extension $\mathbb{A}^{\sigma}$.  Moreover the natural mapping
  $\mu\colon \mathbb{A} \rightarrow \mathbb{\hat A}$ can be extended
  to a surjection $\nu \colon \mathbb{A}^{\sigma} \twoheadrightarrow
  \mathbb{\hat A}$.