Kleisli adjunction of the distribution monad Let $\langle D , \mu, \eta \rangle$ be the distribution monad on $Set$ and let $Kl(D)$ be the Kleisli category on the distribution monad.  I am interested in the adjunction between $Kl(D)$ and $Set$, so the adjoint pair $G: Kl(D) \rightarrow Set$ and $F: Set \rightarrow Kl(D)$, where. In particular, I want to know exactly what the functors are in the adjunction.  I can see here some information on the adjunction.  In particular, we see what the functor $G$ does to morphisms.

The notation is very confusing in this wikipedia article.  I am especially interested in how morphisms in $Kl(D)$ map to functions in $Set$ according to G.
Can each of these equations be explained in gross detail, showing how we go from a probabilistic function to just a function?
Here are some of the details we need to understand these functors:
$$f:X \rightarrow Y$$
$$\mu : T^2 \rightarrow T$$
$$GY_T = TY$$
$$G(f^* : X_T \rightarrow Y_T) = \mu_Y \circ Tf$$
Can we get the notation explained?  I am so confused by this idea that they talk about $Kl(D)$ morphisms like this $X \rightarrow TY$ and then like this $X_T \rightarrow Y_T$.
I translate the two equations into the notation with D as the functor of the monad thusly:
$$GY_D = DY$$
$$G(f^* : X_D \rightarrow Y_D) = \mu_Y \circ Df$$
... and then try to understand this notation:
$$\mu_Y \circ Df$$
Let's start with $Df$.  Is $Df$ a mapping of distributions on $X$ to distributions on $Y$ by defined by the mapping of the set elements according to $f$?
Then take $\mu_Y \circ Df$.  I think $\mu_Y : D^2(Y) \rightarrow D(Y)$ is a function from distributions of distributions on Y to distributions on Y (it is done by expanding out the mixtures and then grouping terms and adding up).  But then, how does $\mu_Y$ act on on $Df$?  How is  $\mu_Y \circ Df$ just a function mapping a set to a set?
After the explanation, can someone give the intuition of how G takes probabilitic functions to functions?
 A: What I will be saying is true of any Kleisli adjunction, not just the one you're interested in.
Given a monad $(T, \mu, \eta)$ over a category $\mathcal{C}$, the Kleisli category $\mathcal{C}_T$ can be presented in two different ways:

*

*The category of objects $A : \mathcal{C}$ and Kleisli arrows $f : A \to T(B)$ (the one mentioned on the wikipedia page),

*The category of free algebras $T(A)$ over the monad, and algebra morphisms $f : T(A) \to T(B)$ between them.

There is a very close relationship between the two presentations, of course (they are equivalent categories, after all), but in my personal opinion it is way easier to understand the (forgetful) functor you call $G : \mathcal{C}_{T} \to \mathcal{C}$ if you present the Kleisli category in the second way. Let me expand a bit:
A free algebra over the monad $(T, \mu, \eta)$ is an object $T(A) : \mathcal{C}$: it is an algebra because the monad multiplication $\mu : T \circ T \to T$ has components $\mu_A : T(T(A)) \to T(A)$ that automatically satisfy the $T$-algebra axioms.
In this context, the functor $G : \mathcal{C}_T \to \mathcal{C}$ just forgets that the objects $T(A)$ are algebras and the morphisms $f$ are algebra morphisms:
$$G(f : T(A) \to T(B)) = f : T(A) \to T(B)$$
In much the same way, the (free) functor $F : \mathcal{C} \to \mathcal{C}_T$ is simply given by the monad $T$ itself:
$$F(f : A \to B) = T(f) : T(A) \to T(B)$$
I suspect you won't be satisfied with this treatment, though: you want it phrased in the language of the wikipedia article, that presents the Kleisli category  in a different way. This is not so hard to do, but it gets a little boring. I will try to give you the gist of it, and you can work out the details yourself.
Let us call the wikipedia presentation of the kleisli category $\mathcal{C}_{T}^{\prime}$: there is an equivalence of categories $\mathcal{C}_{T}^{\prime} \simeq \mathcal{C}_{T}$. Let me give you an explicit description of it:
The category $\mathcal{C}_{T}^{\prime}$ has objects the objects of the base category, but its morphisms $f : A \rightsquigarrow B$ are morphisms $f : A \to T(B)$ in the base category $\mathcal{C}$. How do we recover the corresponding morphisms in $\mathcal{C}_{T}$, you ask? Easy! Just consider their image under the monad $\mathcal{T}$ and compose with the multiplication $\mu$:
$$T(A) \xrightarrow{T(f)} T(T(B)) \xrightarrow{\mu_B} T(B)$$
So, the functor $\mathcal{C}_{T}^{\prime} \to \mathcal{C}_{T}$ sends objects $A$ to objects $T(A)$ and morphisms $f$ to morphisms $\mu \circ T(f)$.
The other way around is just as easy to describe on morphisms, so I will only do that: we want to send a morphism $T(f) : T(A) \to T(B)$ to a morphism $A \to T(B)$. Just as easy as before: just use the monad unit $\eta$!
$$A \xrightarrow{\eta_A} T(A) \xrightarrow{T(f)} T(B)$$
As a final note: if you want to look at how the functor $\mathcal{C}_{T} \to \mathcal{C}_{T}^{\prime}$ acts on objects, remember that you will only be interested in their isomorphism equivalence class (in $\mathcal{C}$): isomorphic objects will give rise to isomorphic objects in the kleisli category anyways, so nothing important is lost this way.
I can (and will) let you work out the details of how to define $G$ and $F$ as adjoint functors between $\mathcal{C}$ and $\mathcal{C}_{T}^{\prime}$, given the equivalence I just sketched and the description of the adjucntion between $\mathcal{C}$ and $\mathcal{C}_{T}$, above.
For the record, all this is spelled out in the n-lab page on Kleisli categories, in (more or less) the same way.
From this, you can produce a more concrete description of $G$ and $F$ for the case of the distribution monad $\mathcal{D}$, since you're interested that specifically. It is important, though, to stress that the construction in this answer works for any monad over any category.
