Yoneda lemma

The Yoneda lemma identifies morphisms from a functor into a representable functor with elements of the functor at the representing object. It implies that an object is determined up to isomorphism by its morphisms from every object.

Representable functor: A functor naturally isomorphic to morphisms into or out of a fixed object. The lemma’s central comparison is between a functor and one represented by an object.
Category theory: The mathematical study of objects, morphisms, and structure-preserving composition. The lemma is a general theorem about objects and morphisms in categories.
Yoneda embedding: The fully faithful functor sending each category object to its representable Hom functor. It packages the lemma as an embedding of a category into a functor category.
Nobuo Yoneda: Japanese mathematician whose work established the lemma now bearing his name. The theorem is named for Yoneda, who introduced it in category theory.
Hom functor: A functor that assigns sets of morphisms to objects and maps between those sets to morphisms. The lemma applies to natural transformations whose target is a Hom functor.
Category: A structure consisting of objects, morphisms, identity maps, and associative composition. The theorem is stated for objects and morphisms belonging to a category.
Universal property: A characterization of an object by the unique maps it admits or receives. The lemma turns many universal properties into statements about natural transformations.
Samuel Eilenberg: Polish-American mathematician who co-founded category theory with Saunders Mac Lane. Eilenberg helped create the framework in which the lemma became a central result.
Natural transformation: A family of maps between functors that commutes with every map in their domain category. The theorem identifies natural transformations into representables with elements.
Functor: A structure-preserving map between categories that sends objects and morphisms to objects and morphisms. The lemma compares functors defined from morphism sets.

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Commutative diagramRelated: Its proof and applications are commonly organized through naturality and commutative diagrams.