Naturality
Naturality is a central idea in category theory describing how a construction behaves uniformly with respect to morphisms. In its standard form, a natural transformation η between two functors F and G from a category C to a category D assigns to each object X in C a morphism η_X: F(X) -> G(X) in D, such that for every morphism f: X -> Y in C, the naturality condition G(f) ∘ η_X = η_Y ∘ F(f) holds. Equivalently, the action of F and G on arrows commutes with the components η_X, making the naturality square commute for each f.
Naturality expresses that the transformation is compatible with the structure of the category and respects morphisms
Variants and extensions include naturality in enriched or higher category theory, where coherence conditions replace simple