Hi Mateusz! Nice to meet someone mathematically-oriented! I'm delighted with your care about the proper use of algebraic language and am always eager to be corrected. However this time there is no need to do so -- it IS a monoid, one of those you meet almost everywhere, i.e. a monoid of X->X transformations. The identity map is it's neutral element (identity, unit) and composition (X->X)x(X->X)->(X->X) is its action. Notice that it is not a group since some transformations cannot be reverted -- e.g. the ones produced with mk-remover. The code is a bit messy and it mostly comes from a one-night hackaton, and I will not be able to polish it until next weekend for sure. I will translate the Polish comments. At the moment I can only provide you with the last one, a neat mixture of Gombrowicz and Maslowska: "koniec bomba a kto czytal ten ssie galy eurocwelom" -- the translation does not seem trivial, perhaps something like "The end and the bomb, who read is an euro-pansy cocksucker" -- however it might look a bit homophobic (as many Polish curses do), perhaps you or Panicz could propose something better?
Greeting, d. PS I just found your answer, so now I guess we do agree on the monoid thing? As of monads I do understand that the class of endofunctors with composition and identity as unit form a monoid, no doubt, that's the construction I mentioned at the beginning. But that's the monad-monoid relation in the direction opposite to what I was asking for. And technically I am not sure whether "the endofunctor definition" applies to Kleisli triplets (which you probably meant) that easily, as it seems either broader, or that the "all told" part hides some important constraints, like that the category C you consider has enough means (perhaps binary products or some other universal constructions) to represent "the new type" T(A) for any A in C... But again, I am an ignorant so please do correct me.
