natural transformation

Once functors are seen as maps between categories, the obvious next question is: what is a map between two functors? A natural transformation is the answer. It is a way of turning one functor F into another functor G uniformly, by choosing a morphism F(A) -> G(A) at every object A, all chosen so coherently that they commute with everything in sight. The word “natural” captures the informal sense that a construction is canonical — it does not depend on arbitrary choices like a basis.

Given functors F, G : C -> D, a natural transformation η : F => G is a family of morphisms η_A : F(A) -> G(A), one for each object A of C, called the components, subject to the naturality condition: for every morphism f : A -> B in C, the square commutes, meaning G(f) ∘ η_A = η_B ∘ F(f). If every component η_A is an isomorphism, η is a natural isomorphism, and F and G are then “the same up to canonical identification.”

Naturality is what distinguishes a deep theorem from a coincidence. A finite-dimensional vector space V is isomorphic to its dual V*, but no isomorphism is natural — it requires choosing a basis. By contrast the map V -> (V*)* into the double dual, sending a vector to evaluation-at-it, IS natural, which is the precise sense in which the double dual is canonically the original space. Functors C -> D and natural transformations between them themselves form a category, the functor category D^C.