Possibilistic operators in Formal Concept Analysis as Kan extensions

In this paper we prove that Dubois--Prade's eight possibilistic operators in Formal Concept Analysis arise canonically from Kan extensions of the underlying boolean profunctor. This provides a conceptual explanation for the result that $N\Pi$-pairs are the formal concepts of the complement context. We further prove that the FCA closure operator and the $N\Pi$-pairs are the only symmetric or asymmetric operator compositions that give formal concepts. Finally we use these eight possibilistic operators to construct new closure operators on a formal context via standard categorical arguments.