Natural Transformation - Yoneda Lemma

Yoneda Lemma

If X is an object of a locally small category C, then the assignment Y ↦ Hom_C(X, Y) defines a covariant functor F_X : C → Set. This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of X). The natural transformations from a representable functor to an arbitrary functor F : C → Set are completely known and easy to describe; this is the content of the Yoneda lemma.