Natural transformation

In category theory, a natural transformation is a systematic way to map one functor onto another, preserving the compositional structure of the categories. Because it acts as a bridge between functors, it is sometimes called a "morphism of functors." Along with categories and functors themselves, it is a core concept in the field.

Formally, given two functors \(F\) and \(G\) that both go from category \(\mathcal{C}\) to category \(\mathcal{D}\), a natural transformation \(\eta\) from \(F\) to \(G\) consists of a family of morphisms \(\eta_X : F(X) o G(X)\), one for each object \(X\) in \(\mathcal{C}\). Each \(\eta_X\) is called the component of \(\eta\) at \(X\). These components must satisfy a consistency condition: for every morphism \(f: X o Y\) in \(\mathcal{C}\), the equation \(\eta_Y \circ F(f) = G(f) \circ \eta_X\) holds, which can be shown as a commutative diagram. If \(F\) and \(G\) are contravariant, the vertical arrows in that diagram are reversed.

We write \(\eta: F o G\) or \(\eta: F \Rightarrow G\) to denote a natural transformation, and we say the family \(\eta_X\) is natural in \(X\). When every component \(\eta_X\) is an isomorphism in \(\mathcal{D}\), the transformation is called a natural isomorphism (or natural equivalence). Two functors are naturally isomorphic if such a transformation exists between them. A weaker notion, an infranatural transformation, is simply the family of components without requiring the naturality condition.