Natural Isomorphism

Idea

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

Definition

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

References

• riehl2016-category-theory