Kleisli and Eilenberg-Moore Constructions as Parts of Biadjoint Situations

We consider monads over varying categories, and by defining the morphisms of Kleisli and of Eilenberg-Moore from a monad to another and the appropriate transformations (2-cells) between morphisms of Kleisli and between morphisms of Eilenberg-Moore, we obtain two 2-categories MndKl and MndEM. Then we prove that MndKl and MndEM are, respectively, 2-isomorphic to the conjugate of Kl and to the transpose of EM, for two suitably defined 2-categories Kl and EM, related, respectively, to the constructions of Kleisli and of Eilenberg-Moore. Next, by considering those morphisms and transformations of monads that are simultaneously of Kleisli and of Eilenberg-Moore, we obtain a 2-category Mndalg, of monads, algebraic morphisms, and algebraic transformations, and, to confirm its naturalness, we, on the one hand, prove that its underlying category can be obtained by applying the Ehresmann-Grothendieck construction to a suitable contravariant functor, and, on the other, we provide an explicit 2-embedding of a certain 2-category, Sigpd, of manysorted signatures (hence also of another 2-category Spfpd, of many-sorted specifications), arising from the field of many-sorted universal algebra, into a 2-category of the type Mndalg. Moreover, we investigate for the adjunctions between varying categories the counterparts of the concepts previously defined for the monads, obtaining several 2-categories of adjunctions, as well as several 2-functors from them to the corresponding 2-categories of monads, and all in such a way that the classical Kleisli and Eilenberg-Moore constructions are left and right biadjoints, respectively, for these 2-functors. Finally, we define a 2-category Adalg, of adjunctions, algebraic squares, and algebraic transformations, and prove that there exists a canonical 2-functor Mdalg from Adalg to Mndalg.