Natural Transformation

Idea

In category theory, a natural transformation is an transformation between pairs of functors. The naturality condition makes it a 2-morphisms in the category Cat.

Definition

Let $\mathcal{C},\mathcal{D}$ be categories. Let $F,G:\mathcal{C}\Rightarrow \mathcal{D}$ be functors. A natural transformation $\alpha :F\Rrightarrow G$ is a family of morphism ${\alpha }_x:\mathcal{C}[Fx,Gx]$ such that, for all pair of objects $x,y\in \mathcal{C}$ such that the following naturality square commutes:

References

• riehl2016-category-theory