The preprint whose abstract follows is available by anonymous ftp from the machine sun1.mta.ca in the directory pub/papers/rosebrugh as ccd4.{tex,dvi}. The source file is LaTeX and also requires the files catmac.sty (for diagram macros) and cat.bib which are found in the same directory. Paper copies are available from Rosebrugh - be sure to include a postal address if requesting one. Constructive complete distributivity IV Robert Rosebrugh and R. J. Wood Abstract A complete lattice L is constructively completely distributive, ccd, when the sup arrow from down-closed subobjects of L to L has a left adjoint. The Karoubian envelope of the (bi-)category of relations is equivalent to the (bi-)category of ccd lattices and sup-preserving arrows. The equivalence restricts to an equivalence between ideals and ``totally algebraic'' lattices. Both equivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to an equivalence with ccd lattices in a topos. ==============================================================================