Category theory MOC

Natural transformation

A natural transformation is a morphism in a so-called functor category, that is it is a morphism between two functors, or a 2-morphism in Category of small categories. If , then a natural transformation consists of a morphism for every such that the following diagram commutes: cat

i.e. for every .¹

If is an isomorphism for every , then it is called a Natural isomorphism and we say .

A slight generalization is an (Extra)natural transformation.

Properties

• The most fundamental result in category theory: the Yoneda lemma
• Identity natural transformation

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Footnotes

1. 2020, Topology: A categorical approach, pp. 11–12 (Definition 0.9) ↩