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