For each monoid (a semigroup with an identity element) $(M,\bullet,e)$ we can define a category $\Kat C(M,\bullet,e)=(\Ob,\Hom,id,\circ)$ with one object by putting $\Ob=\{M\}$ , and morphisms are elements of $M$ : $\Hom(M,M)=M$ , where $id_M=e$ , and the composition $\circ$ of morphisms is the monoidal product $\bullet$ on elements of $M$ : $y\circ x=y\bullet x$ .

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.