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A natural transformation η: F ⇒ G between functors F, G: C → D assigns to each object A in C a morphism η_A: F(A) → G(A) in D such that for every morphism f: A → B in C, the naturality square commutes: η_B ∘ F(f) = G(f) ∘ η_A. Natural transformations are the morphisms between functors, making them the 2-morphisms of the 2-category Cat. The concept of 'naturality' formalizes the intuition that a construction is canonical or independent of arbitrary choices—the determinant, double dual embedding, and many algebraic maps are natural transformations.
Verify that the double dual embedding V → V** for vector spaces (sending v to the evaluation map ev_v) is natural by drawing and checking the naturality square for an arbitrary linear map T: V → W. Contrast with the non-natural isomorphism V ≅ V* (which requires choosing a basis).
You have already seen that a functor F: C → D is a structure-preserving map between categories — it sends objects to objects and morphisms to morphisms while respecting composition and identities. A natural transformation takes the next step: it is a map between two functors that themselves go between the same pair of categories. If F and G are both functors from C to D, a natural transformation η: F ⇒ G gives, for each object A in C, a morphism η_A: F(A) → G(A) in D. These components must fit together coherently across all morphisms of C.
The coherence condition is the naturality square. For any morphism f: A → B in C, the square formed by F(f), G(f), η_A, and η_B must commute: η_B ∘ F(f) = G(f) ∘ η_A. Think of it this way: you can either first apply F to f (getting a morphism in D between the F-images) and then translate from the F-image to the G-image via η_B, or you can first translate at A via η_A and then apply G to f. The naturality condition says these two paths yield the same morphism. This is a genuine constraint — many component-wise maps of the right type fail it.
The canonical example is the double dual embedding. For a finite-dimensional vector space V, define η_V: V → V by η_V(v) = ev_v, where ev_v(φ) = φ(v). This map is defined purely in terms of evaluation — no basis, no inner product, no arbitrary choices. For any linear map T: V → W, the naturality square commutes: T ∘ η_V = η_W ∘ T. Contrast this with the isomorphism V ≅ V* (dual space). Such an isomorphism exists and V and V* have the same dimension, but constructing a specific one requires choosing a basis or an inner product. The resulting map fails the naturality square for arbitrary T because the choice of basis on V and W may not be compatible. The difference is exactly what "natural" means: the double dual map is canonical; the V ≅ V* map is not.
Natural transformations are important partly because they are the morphisms of functor categories: the collection of all functors from C to D forms a category, denoted [C, D] or DC, where objects are functors and morphisms are natural transformations. This is the beginning of higher-dimensional category theory. Natural transformations also compose: given η: F ⇒ G and ε: G ⇒ H, the composite ε ∘ η: F ⇒ H is defined component-wise by (ε ∘ η)_A = ε_A ∘ η_A, and the naturality squares paste together correctly.
The concept of a natural isomorphism — a natural transformation where every component η_A is an isomorphism — formalizes the idea that two functors are "the same up to canonical isomorphism." This is subtly stronger than just knowing F(A) ≅ G(A) for each A separately; a natural isomorphism guarantees those isomorphisms are compatible with all morphisms. Many fundamental equivalences in algebra and topology are natural isomorphisms, and recognizing them as such is often the key to transferring results from one context to another.