What is the action on $1$-cells of the functor sending a monad to its EM adjunction? What about the Kleisli adjunction?
Let $A$ be the walking adjunction. Recall that an adjunction is the same thing as a $2$-functor $$\dashv\ :A\to\mathfrak{Cat},$$ where we use the fully algebraic definition of an adjunction.
Let $M$ be the spinning monad, i.e. a $2$-category with one object $X$, one nontrivial $1$-cell $T:X\to X$ and two nontrivial $2$-cells $\eta:1_X\Rightarrow T$, $\mu:T\circ T\Rightarrow T$ satisfying the monad identities. A monad is just a $2$-functor $$(-,-,-):M\to\mathfrak{Cat}.$$
These identifications naturally give us two $2$-cateogries $${\bf Adj}=\mathfrak{Cat}^A,$$ $${\bf Mon}=\mathfrak{Cat}^M,$$ whose objects are adjunctions and monads, respectively. The notion of a pseudonatural transformation between $2$-functors yields notions of morphism between adjunctions and monads under this identification, unpacked below.
The nlab page on the $2$-category of adjunctions in a bicategory points out that $M$ is isomorphic to the full subcategory of $A$ on the 'left object', and that this naturally induces a 'structure' $2$-functor $${\bf Str}:{\bf Adj}\to {\bf Mon}$$ sending an adjunction $F\dashv G:\mathcal{C}\rightleftarrows\mathcal{D}$ to the monad $(G\circ F,\eta,1_G\star\epsilon\star1_F)$ on $\mathcal{C}$. The page then states that the EM and Kleisli constructions are 'the adjoints' to this functor (presumably with the EM functor as right adjoint and Kleisli as left so that comparison functors form the components of the respective unit/counit), with the reference being a 3-page note written by Schanuel and Street. This note in turn references the famous 1958 paper of Kan as the source for how to construct these left and right adjoints using Kan extensions, however after scanning the paper I was unable to find the relevant section (very possible that I just missed it due to unfamiliarity with Kan extensions).
User varkor was kind enough to point me in the direction of Linton's An Outline of Functorial Semantics for a general perspective on structure-semantics adjunctions, but I was unable to find this particular case listed in his paper (again very possible that I just missed it). They also pointed me towards Pumplün's Eine Bemerkung über Monaden und adjungierte Funktoren as a source for this particular result, but I don't have access to the article and my wife isn't fluent enough in German to translate it for me yet no matter how many times I bug her.
Do any categorically literate users here have access to the journal and the ability to translate it, or can anyone reproduce the result?
Here are the unpacked definitions of 'morphism of adjunctions' and 'morphism of monads' mentioned above.
${\bf Definition.}$ Let $F\dashv G:\mathcal{C}\rightleftarrows\mathcal{D}$ and $F'\dashv G':\mathcal{C}'\rightleftarrows\mathcal{D}'$ be adjunctions. A morphism of adjunctions from $F\dashv G$ to $F'\dashv G'$ consists of a quadruplet $$(A,B,\digamma,\Gamma)$$ where $A:\mathcal{C}\to\mathcal{C}'$ and $B:\mathcal{D}\to\mathcal{D}'$ are functors, and
are natural isomorphisms, such that the following natural transformation diagram commutes in ${\bf Fun}$
The coherence condition above is equivalent to a condition on the counit and to a certain hom-class diagram commuting, in similar fashion to how adjunctions can be formulated in terms of units, counits, or a condition on hom-classes.
${\bf Definition.}$ Let $T:\mathcal{C}\to\mathcal{C}$ and $T':\mathcal{C}'\to\mathcal{C}'$ be monads. A morphism of monads from $T$ to $T'$ consists of an ordered pair $$(F,\alpha)$$ where $F:\mathcal{C}\to\mathcal{C}'$ is a functor and
is a natural isomorphism, such that the following natural transformation diagrams commute in ${\bf Fun}$
Given a morphism of adjunctions $(A,B,\digamma,\Gamma):F\dashv G\longrightarrow F'\dashv G'$, we naturally obtain a morphism of the corresponding monads $$\big(A,(1\star\digamma)\circ(\Gamma\star1)\big):G\circ F\longrightarrow G'\circ F'.$$ Given a morphism of monads, however, I don't see how to construct a morphism between the corresponding EM adjunctions. User varkor pointed out that the notion of morphism of adjunctions that Pumplün uses is weaker than the one given above (presumably coming from lax natural transformations between $2$-functors, so $\digamma$ and $\Gamma$ are no longer necessarily invertible), however there still appear to be issues even in this case. Any relevant references or thoughts are appreciated.
