This appears as remark 4.3 in (Caramello 2011).
A complete lattice is called constructively completely distributive (CCD) if the join-assigning morphism has a left adjoint, with the poset of downsets.
Completely distributive lattices correspond to tight Galois connections (Raney 1953). This generalizes to a correspondence between totally distributive toposes and essential localizations (Lucyshyn-Wright 2011).
CCD lattices are precisely the nuclear objects in the category of complete lattices.
The (bi-) category with CCD lattices and sup-preserving maps is the idempotent splitting of the (bi-) category of relations . This plays an important role in domain-theoretical approaches to the semantics of linear logic.
R. Guitart, J. Riguet, Enveloppe Karoubienne de Catégories de Kleisli , Cah. Top. Geom. Diff. Cat. XXXIII no.3 (1992) pp.261-266. (pdf)
G. N. Raney, Tight Galois Connections and Complete Distributivity , Trans.Amer.Math.Soc 97 (1960) pp.418-426. (pdf)
R. Rosebrugh, R. J. Wood, Constructive complete distributivity IV , App. Cat. Struc. 2 (1994) pp.119-144. (preprint)
I. Stubbe, Towards “Dynamic Domains”: Totally Continuous Complete Q-Categories , Theo. Comp. Sci. 373 no.1-2 (2007) pp.142-160.