The left part of a pair of adjoint functors is one of two best approximations to a weak inverse of the other functor of the pair. (The other best approximation is the functor's right adjoint, if it exists. ) Note that a weak inverse itself, if it exists, must be a left adjoint, forming an adjoint equivalence.
A left adjoint to a forgetful functor is called a free functor; in general, left adjoints may be thought of as being defined freely, consisting of anything that an inverse might want, regardless of whether it works.
The concept generalises immediately to enriched categories and in 2-categories.
Given posets (or prosets) and and a monotone function , a left adjoint of is a monotone function such that
for all in and in .
Given locally small categories and and a functor , a left adjoint of is a functor with a natural isomorphism between the hom-set functors
Given -enriched categories and and a -enriched functor , a left adjoint of is a -enriched functor with a -enriched natural isomorphism between the hom-object functors
Given categories and and a functor , a left adjoint of is a functor with natural transformations
(where etc gives the composite in the forwards, anti-Leibniz order) satisfying certain triangle identities.
Given a 2-category , objects and of , and a morphism in , a left adjoint of is a morphism with -morphisms
satisfying the triangle identities.
Although it may not be immediately obvious, these definitions are all compatible.
Whenever is a left adjoint of , we have that is a right adjoint of .
Left adjoint functors preserve