Given a monad, $(M, \mu, \eta)$, where $M: C \rightarrow C$ for some category $C$, there is a category of factorizations, $F\cdot G = M$ where $F: X \rightarrow C$, $G: C \rightarrow X$. Though this may be a rather open question, does anyone have any examples of calculations whose result is the computation of some particular factorization that has some desired property? What are the methods of calculation? The result of the computation should be a category and the pair $F, G$. What might be an algorithm given a monad on a finite category that computes these objects and functors?


I will assume that G⊣F, And that the maps between factorizations are functors that commute nicely with the units and counits.


As per a comment, I would like to focus on the Kleisli category. With that said, I believe the comment has suggested that there is no way to have a "means" or requirement that would cause you to calculate anything else in the category of factorizations, aside from the category of M-algebras (which is Eilenberg-Moore?). This seems like a question all its own.



Computing a factorization of a monad
Your last question, the one you call "a question all its own," is essentially the subject of [this paper](https://link.springer.com/article/10.1007%2Fs10485-016-9473-8). (Free version from the author's website [here](http://www.math.wayne.edu/~asalch/papers/kleisli6c.pdf).)


Here is a relevant excerpt from the introduction. The notation is as follows: $T$ is a monad, $C^T$ is the Eilenberg-Moore category of that monad (i.e., the category of $T$-algebras), and a "homological presentation of $T$" is an adjunction whose associated monad is $T$ and which satisfies a certain condition defined in Definition 2.2 in the paper. The relevant excerpt: 



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> In Theorem 2.20, when $T$ is coequalizable and $\mathcal{C}^T$ has a biproduct and is Krull-Schmidt, we actually construct a "coordinate system" on the natural equivalence classes of homological presentations of $T$! Any homological presentation is determined uniquely (up to natural equivalence) by specifying a suitable subcollection of the collection of isomorphism classes of indecomposable objects of $\mathcal{C}^T$. In some practical cases, $\mathcal{C}^T$ is the category of finitely generated modules over an algebra, and then the vertices of the Auslander-Reiten quiver of $\mathcal{C}^T$ act as "coordinates" for the collection of natural equivalence classes of homological presentations of $T$.
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A fairly explicit example is given in section 4, computing all the "finitary" homological presentations of the free $A$-module monad on the category of sets, where $A$ is a Dedekind domain.


