A functor from a category to itself is called an endofunctor.
Given any category , the functor category
is called the endofunctor category of . The objects of are endofunctors , and the morphisms are natural transformation between such endofunctors.
The endofunctor category is a strict monoidal category, thanks to our ability to compose endofunctors:
The unit object of this monoidal category is the identity functor from to itself:
A monoid in this endofunctor category is called a monad on .