  In a recent paper, Amini et al. introduce a general framework to prove
duality theorems between special decompositions and their dual combinatorial
object. They thus unify all known ad-hoc proofs in one single theorem. While
this unification process is definitely good, their main theorem remains quite
technical and does not give a real insight of why some decompositions admit
dual objects and why others do not. The goal of this paper is both to
generalise a little this framework and to give an enlightening simple proof of
its central theorem.
