Computing a factorization of a monad

Question

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 choice of some particular factorization that has some desired property? What are the methods of calculation? So far, I know that the Kleisli category is initial in the category of factorizations, while Eilenberg Moore is terminal. SOmetimes initial categories are given the symbol 0, terminal likewise 1. Does this suggest some kind of arithmetic or calculational method? I am picturing a calculation that is done in the category of factorizations.