Now the covariant hom-functor corresponds to the action of on itself by left-multiplication (the contravariant version corresponds to right-multiplication). The Yoneda lemma with states that , that is, the equivariant maps from this -set to itself are in bijection with . But it is easy to see that (1) these maps form a group under composition, which is a subgroup of , and (2) the function which gives the bijection is a group homomorphism. (Going in the reverse direction, it associates to every in the equivariant map of right-multiplication by .) Thus is isomorphic to a subgroup of , which is the statement of Cayley's theorem.
now the covariant hom-functor corresponds to the action of on itself by left-multiplication (the contravariant version corresponds to right-multiplication) the yoneda lemma with states that  that is the equivariant maps from this -set to itself are in bijection with  but it is easy to see that (1) these maps form a group under composition which is a subgroup of  and (2) the function which gives the bijection is a group homomorphism (going in the reverse direction it associates to every in the equivariant map of right-multiplication by ) thus is isomorphic to a subgroup of  which is the statement of cayley's theorem