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In this article, the author analyses distributive and mixed distributive lawsand some of their equivalences through the use of 2-adjunctions of the type$\Adj$-$\Mnd$. As far as the distributive laws are concerned, the equivalencebetween this structures and monads objects in the 2-category $\Adj_{\R}(Cat)$is analysed, where these monad objects will correspond to liftings to thecategory of algebras of Eilenberg-Moore. Second, the equivalence between thesestructures and a pair consisting of a Eilenberg-Moore lifting and a Kleisliextension is analysed too, according to E. Manes and P. Mulry (2010), where theauthor was able to recast the theorem with an additional naturality on theinvolved monads. On the other hand, the equivalence between mixed distributivelaws and comonad objects in the same 2-category $\Adj_{\R}(Cat)$ was analysedalso within the context of a 2-adjunction. This comonad object will correspondto a comonad lifting structure on the category of algebras of Eilenberg-Moore.Finally, a similar theorem relating mixed distributive laws with a pair ofEilenberg-Moore liftings on algebras and coalgebras is proved, with anaturality for the equivalence on the monad and comonad involved. The samestated objective in a previous installment, by the author and company, isfollowed. That is to say, using 2-adjunctions to analyse classical monad theoryin order to provide clarity in the proofs and naturality on the equivalences.

Monad and Comonad Objects through 2-adjunctions of the type Adj-Mnd