is called the endofunctor category of $C$. The objects of $End(C)$ are endofunctors $F: C \to C$, and the morphisms are natural transformation between such endofunctors.

Properties

Monoidal structure

The endofunctor category is a strictmonoidal category, thanks to our ability to compose endofunctors:

$\circ : End(C) \times End(C) \to End(C)$

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

$1_C \in End(C)$

Monoids

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