End_C(x) = Hom_C(x,x) ,
which may be written if the category 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 be a diagram in , where is a monoidal category (in the case above the monoidal structure is the cartesian product and is a constant diagram from the initial category). One defines as an object in , equipped with a natural transformation which is universal in the sense that for all objects , and any natural transformation there exists a unique morphism such .
If the universal object exists then there is a unique structure of a monoid , such that the map is an action.