Functor

A functor between categories sends each object to an object and each morphism to a morphism , preserving composition and identities:

A functor is a structure-preserving map of categories, acting compatibly on objects and on arrows. A functor may be faithful, full, or essentially surjective according to how much it forgets, which corresponds to property, structure, and stuff.

Variance and presheaves

A functor out of the opposite category, , is contravariant and reverses the direction of morphisms. A contravariant functor into is a presheaf; the covariant one is a copresheaf, the representable copresheaf being .

Natural transformation

A natural transformation between functors is a family of morphisms making the naturality square commute for every : . Functors and their natural transformations form a functor category.

References

• Eilenberg & Mac Lane, “General Theory of Natural Equivalences”, Trans. AMS (1945)
• Saunders Mac Lane, Categories for the Working Mathematician (1971; 2nd ed. 1998)