An equational approach to enriched distributivity

The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of "truth values"), via an appropriate distributive law between the "down-set" monad and the "up-set" monad on the category of quantale-enriched categories. If the underlying lattice of the quantale is completely distributive, and if powers distribute over non-empty joins in the quantale, then this distributive law can be concretely formulated in terms of operations, equations and choice functions, similar to the familiar distributive law of lattices.