In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define socalled functor categories. Natural transformations are, after categories and functors, one of the most basic notions of category theory and consequently appear in the majority of its applications.
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Definition
If F and G are functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism η_{X} : F(X) → G(X) in D called the component of η at X, such that for every morphism f : X → Y in C we have:
This equation can conveniently be expressed by the commutative diagram
If both F and G are contravariant, the horizontal arrows in this diagram are reversed. If η is a natural transformation from F to G, we also write η : F → G or η : F ⇒ G. This is also expressed by saying the family of morphisms η_{X} : F(X) → G(X) is natural in X.
If, for every object X in C, the morphism η_{X} is an isomorphism in D, then η is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G.
An infranatural transformation η from F to G is simply a family of morphisms η_{X}: F(X) → G(X). Thus a natural transformation is an infranatural transformation for which η_{Y} o F(f) = G(f) o η_{X} for every morphism f : X → Y. The naturalizer of η, nat(η), is the largest subcategory of C containing all the objects of C on which η restricts to a natural transformation.
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