A factorization of regularity

Earlier, the authors showed that categories with regular factorizations (in the sense of Kelly) and with regular epimorphisms closed under composition, are algebras for a KZ-doctrine image on the 2-category catker, of all categories and functors which preserve kernel inclusions. Here we show that the usual 2-category lex, of categories with finite limits and left exact functors, is image the 2-category of algebras for a coKZ-doctrine image on catker. Combined with our earlier results this shows that the 2-category reg, of regular categories and regular functors, is the 2-category of algebras for a distributive law image over the 2-category catker.