Morphisms between functors and the naturality square
If functors are maps between categories, natural transformations are maps between functors. Saunders Mac Lane, co-inventor of category theory, said: "I didn't invent categories to study functors; I invented them to study natural transformations."
A natural transformation α : F ⇒ G assigns to each object A a morphism α_A : F(A) → G(A) such that the naturality square commutes. Watch both paths arrive at the same place!
Key insight: The naturality condition says: α_B ∘ F(f) = G(f) ∘ α_A for every morphism f. Both paths through the square give the same result — that's what makes the transformation "natural" (not dependent on arbitrary choices).
The determinant det : GL_n ⇒ (−)* is a natural transformation between functors on commutative rings! Each component is a group homomorphism — that is the identity det(AB) = det(A) · det(B) — and naturality says det commutes with every ring homomorphism φ: det(φ(A)) = φ(det(A)). Verify the homomorphism identity with concrete matrices below.
Key insight: The determinant is "natural" because it works uniformly over every coefficient ring: applying a ring homomorphism and then taking det gives the same answer as taking det first. This is a concrete example of the naturality square commuting: the two paths always give the same answer.
Functors themselves form a category! In the functor category [C, D], objects are functors F, G, H : C → D and morphisms are natural transformations between them.
Key insight: Natural transformations compose component-wise: (β∘α)_A = β_A ∘ α_A. This gives us a whole new category whose "objects" are functors! This is one of the most powerful ideas in category theory.
Natural transformations appear throughout mathematics. Here are some famous examples that show why "naturality" is such a powerful concept.
The canonical embedding of a vector space into its double dual. Natural because it doesn't require choosing a basis.
Key insight: The common thread: all these transformations work "uniformly" — they don't depend on arbitrary choices. Eilenberg and Mac Lane invented category theory precisely to make the word "natural" mathematically precise!