Natural Transformation

A family of morphisms (called components) η_X : F(X) → G(X) indexed by objects X of a source category, connecting two functors F,G : C → D, such that for every morphism f : X → Y in C the square G(f) ∘ η_X = η_Y ∘ F(f) commutes. It expresses a coherent, objectwise map between functors.

A family of morphisms between two functors with the same domain and codomain, indexed by the objects of the domain category, such that for every morphism in the domain a naturality square commutes; it provides a canonical, structure-preserving comparison between functorial interpretations.

A family of morphisms between two functors F and G from the same source category C to the same target category D: for each object X of C a component η_X : F(X) → G(X) so that for every morphism f:X→Y the square G(f) ∘ η_X = η_Y ∘ F(f) (the naturality condition) commutes.