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.
for all in and in .
Given categories and and a functor , a left adjoint of is a functor with natural transformations
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
See Galois connection for left adjoints of monotone functions.
See adjoint functor for left adjoints of functors.
See adjunction for left adjoints in -categories.
See examples of adjoint functors for examples.