For each monoid (a semigroup with an identity element) we can define a category with one object by putting , and morphisms are elements of : , where , and the composition of morphisms is the monoidal product on elements of : .

Moreover, any category with a single object has a natural structure as a monoid with the binary operation given by the law of composition of morphisms.

Remark. If a monoid is a group, then the identified category again has one object, and furthermore all of its morphisms are isomorphisms. Conversely, a category with one object all of whose morphisms are isomorphisms has a natural structure as a group.