where is the group of permutations of ; in other words, is a G-set. A natural transformation between such functors is the same thing as an equivariant map between -sets: a set function with the property that for all in and in . (On the left side of this equation, the denotes the action of on , and on the right side the action on .) 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. History Yoshiki Kinoshita stated in 1996 that the term "Yoneda lemma" was coined by Saunders Mac Lane following an interview he had with Yoneda in the Gare du Nord station. See also Representation theorem Notes References . External links Mizar system proof: Representable functors Lemmas in category theory Articles containing proofs