* Any discrete group G can be regarded as a one-object groupoid in which case a covariant Set-valued functor is just a G-set. The unique represented functor is the G-set G, with its translation (left multiplication) action. By contrast, a *representable* functor X, not yet equipped with the natural ($G$-equivariant) isomorphism $G \cong X$ defining the representation, is a $G$-torsor.
Related to this is the use of what is essentially the Yoneda lemma in normalisation proofs. For example Britton's lemma (giving normal forms for words in an HNN extension) proceeds by taking the set of "normal forms" and showing that it is a torsor for the group G at issue. In other words one takes a presheaf on G and shows that it is representable, and so is isomorphic to G itself. Another argument of a similar Yoneda kind is the usual proof of the Poincare--Birkhoff--Witt theorem (giving a basis for the universal enveloping algebra of a Lie algebra). In fact the technique is very widely applicable (and very widely applied); for instance you could prove the normal form theorem for the simplicial category using it. A related (but more general) technique is what is sometimes called "normalization by evaluation" by computer scientists; Peter Dybjer has some articles on the relation to the Yoneda lemma. But that is maybe straying a bit far from undergraduate mathematics. Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]