Table of contents |

2 Examples 3 Yoneda lemma 4 Functor categories 5 Historical Notes |

If *K* is a field, then for every vector space *V* over *K* we have a natural" injective linear map *V* `->` *V*^{**} from he vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps form a natural transformation from the identity functor to the double dual functor.

Consider the category **Ab** of abelian groups and group homomorphisms. For all abelian groups *X*, *Y* and *Z* we have a group isomorphism

- Hom(
*X*, Hom(*Y*,*Z*))`->`Hom(*X**Y*,*Z*).

If *X* is an object of the category *C*, then the assignment *Y* `|->` Mor_{C}(*X*, *Y*) defines a covariant functor *F*_{X} : *C* `->` **Set**. This functor is called *representable*. 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.

If *C* is any category and *I* is a small category, we can form the functor category *C ^{I}* having as objects all functors from

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.

The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly, and those of the singular theory, would be isomorphic. But that in itself stated much less than the existence of a natural transformation of the corresponding homology functors.