Adjoint functors

An adjunction is a pair of functors between categories whose hom-sets are naturally isomorphic: maps from one functor’s outputs correspond to maps into the other’s inputs.

Hom-set: A set of morphisms between a fixed pair of objects in a category. Adjunctions compare hom-sets on opposite sides of their defining correspondence.
Category theory: The mathematical study of objects, morphisms, and structure-preserving maps between them. Adjunctions are defined using categories, functors, and morphisms.
Free group: A group generated by a set with no relations beyond the group axioms. The free-group construction is left adjoint to the functor that forgets group structure.
Equivalence of categories: A pair of functors exhibiting two categories as equivalent up to isomorphism of objects. An equivalence is a particularly strong adjunction whose unit and counit are isomorphisms.
Natural transformation: A family of morphisms between functors that commutes with every morphism in their source category. The hom-set bijections must vary compatibly across objects and morphisms.
Functor: A structure-preserving map between categories that sends objects and morphisms to objects and morphisms. Each adjunction consists of two functors acting in opposite directions.
Tensor-hom adjunction: The natural correspondence between linear maps out of a tensor product and linear maps into a hom space. It is a fundamental adjunction in algebra and enriched category theory.
Isomorphism of categories: A pair of inverse functors whose composites are exactly identity functors. Adjoint categories need not be isomorphic or even equivalent.
Unit of an adjunction: The natural transformation from the identity functor on the left category to the right-then-left composite. It packages the adjunction’s canonical maps from objects toward the composite.
Opposite category: A category formed by reversing the direction of every morphism in a category. Adjunctions can be expressed as functors out of a product involving an opposite category.

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