A completely distributive lattice is a
This appears as (Caramello, remark 4.3).
A complete lattice is called constructive completely distributive if the join-assigning morphism , with the poset of downsets. This is equivalent to complete distributivity if and only if the axiom of choice holds; see (WoodFawcett). Constructive completely distributive lattices are an example of continuous algebras for a lax-idempotent 2-monad.
wikipedia, completely distributive lattice
Olivia Caramello, A topos-theoretic approach to Stone-type dualities (arXiv:1103.3493)