All told, a monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of endofunctors and unit set by the identity endofunctor.
At the same time a Monoid is a category with one object. Given a Monoid in the category of endofunctors of X as above, how do we get a category with one object from there? Please specify exactly what the object and the morphisms of this category are.
Followup question: Elements of the Monoid in the category of endofunctors