natural transformation
This term is a technical cornerstone of category theory, describing a way of transforming one functor into another without altering the underlying structure of the categories involved. It is characterized by the requirement that the transformation be "natural," meaning it commutes with the morphisms of the category, ensuring consistency across all objects.
In a mathematical context, this is not a "transformation" in the sense of a geometric rotation or translation, but rather a systemic mapping between functors. It is a highly specialized term used almost exclusively in advanced mathematics, theoretical computer science, and functional programming to describe polymorphic functions or generic mappings.
Meanings
A morphism between two functors that provides a consistent way of transforming one functor into another while preserving the internal structure of the category. It consists of a family of morphisms, one for each object in the domain category, such that for any morphism between objects, a specific square diagram commutes.
The identity functor is a natural transformation from a functor to itself.