Natural transformation

A natural transformation is a family of morphisms connecting two functors with the same source and target categories. Its components commute with every morphism in the source category.

Naturality square: A commutative square expressing compatibility between functorial maps and a natural transformation’s components. Each square states the condition that makes the component family natural.
Category (mathematics): A structure consisting of objects, morphisms, identity morphisms, and associative composition. The source and target categories supply the objects and arrows used by the transformation.
Yoneda lemma: A theorem identifying natural transformations from a representable functor with elements of the represented set or object. Its naturality condition makes representable functors encode an object’s relationships to all others.
Natural isomorphism: A natural transformation whose every component is an isomorphism. It strengthens naturality with componentwise invertibility.
Samuel Eilenberg: A Polish-American mathematician who helped found category theory and algebraic topology. Eilenberg co-developed category theory, including its language for transformations between functors.
Component (category theory): A morphism assigned to each object by a natural transformation between functors. These objectwise morphisms form the family whose compatibility naturality requires.
Morphism: A structure-preserving map between objects in a mathematical setting. Every component of a natural transformation is a morphism in the target category.
Adjunction: A pair of functors related by a natural bijection between two hom-sets. The defining hom-set correspondence must be natural in both variables.
Natural equivalence: A natural transformation used as part of a categorical equivalence, typically with an inverse up to natural isomorphism. It describes equivalence-level sameness rather than an arbitrary transformation.
Saunders Mac Lane: An American mathematician who co-founded category theory and shaped its modern foundations. Mac Lane developed the framework in which natural transformations became a central structural notion.

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