adjunction

Adjunctions are a central concept in category theory that formalize a close relationship between two functors. Given categories C and D, an adjunction consists of functors F: C → D and G: D → C together with a natural isomorphism Hom_D(FX, Y) ≅ Hom_C(X, GY) that is natural in X ∈ C and Y ∈ D. When such an adjunction exists, F is called the left adjoint and G the right adjoint, often written F ⊣ G.

Equivalently, adjunctions can be described by unit and counit natural transformations. There exist η: Id_C → G F (the unit) and ε: F G → Id_D (the counit) such that, for all X in C, ε_{F X} ∘ F(η_X) = id_{F X}, and for all Y in D, G(ε_Y) ∘ η_{G Y} = id_{G Y}. These triangle identities ensure the correspondence between the hom-set isomorphism and the unit–counit data.

Common examples illustrate the idea. The free-forgetful adjunction between sets and groups uses the free group functor F: Set → Grp and the forgetful functor U: Grp → Set, with Hom_Grp(FX, G) ≅ Hom_Set(X, UG). In cartesian closed categories, the product with A is left adjoint to the exponential functor (−)^A, giving natural isomorphisms Hom(B × A, C) ≅ Hom(B, C^A).

Adjunctions preserve and reflect (co)limits: left adjoints preserve colimits and right adjoints preserve limits. If the unit and counit are isomorphisms, the adjunction is an equivalence of categories. A full subcategory with an inclusion functor that has a left (or right) adjoint is called reflective (or coreflective).