Adjunction

A datum of a pair of functors L : C → D and R : D → C together with, for each X in C and Y in D, a natural bijection Hom_D(L(X),Y) ≅ Hom_C(X,R(Y)) that is natural in X and Y. Equivalently specified by unit and counit natural transformations satisfying triangle identities. It encodes a universal best‑approximation relationship: L is left adjoint to R and R right adjoint to L.

A relationship between two functors L : C → D and R : D → C given by a natural bijection Hom_D(L(X), Y) ≅ Hom_C(X, R(Y)) natural in X∈C and Y∈D, equivalently specified by a unit η: Id_C ⇒ R∘L and counit ε: L∘R ⇒ Id_D satisfying the triangle identities.