then the assignment defines a covariant functor . This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of ). The natural transformations from a representable functor to an arbitrary functor are completely known and easy to describe; this is the content of the Yoneda lemma. Historical notes 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 would be isomorphic to those of the singular theory. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups. See also Extranatural transformation Universal property Higher category theory Notes References . External links nLab, a wiki project on mathematics, physics and philosophy with emphasis on the n-categorical point of view André Joyal, CatLab, a wiki project dedicated to the exposition of categorical mathematics formal introduction to category theory. J. Adamek, H. Herrlich, G. Strecker, Abstract and Concrete Categories-The Joy of Cats Stanford Encyclopedia of Philosophy: "Category Theory"—by Jean-Pierre Marquis. Extensive bibliography. List of academic conferences on category theory Baez, John, 1996,"The Tale of n-categories." An informal introduction to higher categories. WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms, categories, functors, natural transformations, universal properties. The catsters, a YouTube channel about category theory. Video archive of recorded talks relevant to categories, logic and the foundations of physics. Interactive Web page which generates examples of categorical constructions in the category of finite sets. Functors