applications of the kleisli and eilenberg-moore 2-adjunctions

In 2010, j. climent vidal and j. soliveres tur developed, among other things, a pair of 2-adjunctions between the 2-category of adjunctions and the 2-category of monads. one is related to the kleisli adjunction and the other to the eilenberg-moore adjunction for a given monad. since any 2-adjunction induces certain natural isomorphisms of categories, these can be used to classify bijections and isomorphisms for certain structures in monad theory. in particular, one important example of a structure, lying in the 2-category of adjunctions, where this procedure can be applied to is that of a lifting. therefore, a lifting can be characterized by the associated monad structure, lying in the 2-category of monads, through the respective 2-adjunction. the same can be said for kleisli extensions. several authors have been discovered this type of bijections and isomorphisms but these pair of 2-adjunctions can collect them all at once with an extra property, that of naturality.