nLab
endofunctor

Definition

A functor from a category to itself is called an endofunctor.

Given any category $C$, the functor category

$$\mathrm{End}(C)=C^C$$End(C) = C^C

is called the endofunctor category of $C$. The objects of $\mathrm{End}(C)$ are endofunctors $F:C\to C$, and the morphisms are natural transformations between such endofunctors. The endofunctor category is a strict monoidal category, thanks to our ability to compose endofunctors:

$$\circ :\mathrm{End}(C)\times \mathrm{End}(C)\to \mathrm{End}(C)$$\circ : End(C) \times End(C) \to End(C)

The unit object of this monoidal category is the identity functor from $C$ to itself:

$$1_C\in \mathrm{End}(C)$$1_C \in End(C)

A monoid in this endofunctor category is called a monad on $C$.