Natural Transformation

category-theory

Definition

Natural Transformation

Let and be categories and let be two functors. A natural transformation assigns to every object a morphism

in such that for every morphism in the following naturality square commutes:

That is, .

Naturality

The equation is the naturality condition. It states that moving from to along either path of the square yields the same result. In other words, the transformation is natural precisely when it is compatible with the action of both functors on morphisms.