Natural transformation

Summary

Let $F, G \colon \mathcal{C} \to \mathcal{D}$ be two functors. A natural transformation $\alpha \colon F \Rightarrow G$ is a family of morphisms $\alpha_A \colon F(A) \to G(A)$, one for each object $A$. The family must commute with the images of all morphisms of $\mathcal{C}$. It makes precise the idea of a construction that is "natural", that is, defined in the same way for all objects without arbitrary choices. The canonical map from a vector space to its double dual is natural. By contrast, an isomorphism of a finite-dimensional space with its dual depends on a choice of basis. Natural transformations are the morphisms of functor categories and enter the definitions of equivalence of categories and of adjoint functors.