The right adjoint functor of a functor, if it exists, is one of two best approximations to a weak inverse of that functor. (The other best approximation is the functor's left adjoint, if it exists.) Note that a weak inverse itself, if it exists, must be a right adjoint, forming an adjoint equivalence.
A right adjoint to a forgetful functor is called a cofree functor; in general, right adjoints may be thought of as being defined cofreely, consisting of anything that works in an inverse, regardless of whether it’s needed.
for all in and in .
Given categories and and a functor , a right adjoint of is a functor with natural transformations
satisfying certain triangle identities.
Given a 2-category , objects and of , and a morphism in , a right 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 right adjoint of , we have that is a left adjoint of .
Right adjoint functors preserve
See Galois connection for right adjoints of monotone functions.
See adjoint functor for right adjoints of functors.
See adjunction for right adjoints in -categories.
See examples of adjoint functors for examples.