An adjunction between functors and , with left adjoint to , is a natural isomorphism of hom-sets . Equivalently it is a unit and counit obeying the triangle identities and . Left adjoints preserve colimits and right adjoints preserve limits, while the composite is a monad and a comonad. The prototypes are free forgetful, taking a set-and-function to the free group, module, or vector-space on it; the currying adjunction ; and , the sum, base change, and product along a map. An adjunction pins down a construction together with its best one-sided approximation, so it is the universal property made functorial.
References
Daniel M. Kan, “Adjoint Functors”, Trans. AMS (1958)
Saunders Mac Lane, Categories for the Working Mathematician (1971), ch. IV