Given an object $x$, the endomorphisms of $x$ form a monoid under composition, the endomorphism monoid of $x$:

$End_C(x) = Hom_C(x,x)
,$

which may be written $End(x)$ if the category $C$ is understood. Up to equivalence, every monoid is an endomorphism monoid; see delooping.

An endomorphism monoid is a special case of a monoid structure on an end construction. Let $d:D\to C$ be a diagram in $C$, where $C$ is a monoidal category (in the case above the monoidal structure is the cartesian product and $d$ is a constant diagram from the initial category). One defines $End(d)$ as an object in $C$, equipped with a natural transformation $a: End(d) \otimes d \to d$ which is universal in the sense that for all objects $Z \in C$, and any natural transformation $f: Z \otimes d \to d$ there exists a unique morphism $g: Z \to End(d)$ such $a \circ (g \otimes d) = f: Z \otimes d \to d$.

Proposition

If the universal object $(End(d),a)$ exists then there is a unique structure of a monoid $\mu: End(d) \otimes End(d) \to End(d)$, such that the map $a: End(d) \otimes d \to d$ is an action.