Editing Adjoint functors (section)

== Properties ==

===Existence===
{{See also|Formal criteria for adjoint functors}}
{{anchor|Freyd's adjoint functor theorem}}Not every functor ''G'' : ''C'' → ''D'' admits a left adjoint. If ''C'' is a [[complete category]], then the functors with left adjoints can be characterized by the '''adjoint functor theorem''' of [[Peter J. Freyd]]: ''G'' has a left adjoint if and only if it is [[limit (category theory)#Preservation of limits|continuous]] and a certain smallness condition is satisfied: for every object ''Y'' of ''D'' there exists a family of morphisms

:''f''<sub>''i''</sub> : ''Y'' → ''G''(''X''<sub>''i''</sub>)

where the indices ''i'' come from a ''set'' {{mvar|I}}, not a ''[[class (set theory)|proper class]]'', such that every morphism

:''h'' : ''Y'' → ''G''(''X'')

can be written as

:''h'' = ''G''(''t'') <math>\circ</math> ''f''<sub>''i''</sub>

for some ''i'' in {{mvar|I}} and some morphism

:''t'' : ''X''<sub>''i''</sub> → ''X'' &isin; ''C''.

An analogous statement characterizes those functors with a right adjoint.

An important special case is that of [[locally presentable category|locally presentable categories]]. If <math>F : C \to D</math> is a functor between locally presentable categories, then

* ''F'' has a right adjoint if and only if ''F'' preserves small colimits
* ''F'' has a left adjoint if and only if ''F'' preserves small limits and is an [[accessible functor]]

===Uniqueness===

If the functor ''F'' : ''D'' → ''C'' has two right adjoints ''G'' and ''{{prime|G}}'', then ''G'' and ''{{prime|G}}'' are [[natural transformation|naturally isomorphic]]. The same is true for left adjoints.

Conversely, if ''F'' is left adjoint to ''G'', and ''G'' is naturally isomorphic to ''{{prime|G}}'' then ''F'' is also left adjoint to ''{{prime|G}}''. More generally, if 〈''F'', ''G'', ε, η〉 is an adjunction (with counit–unit (ε,η)) and
:σ : ''F'' → ''{{prime|F}}''
:τ : ''G'' → ''{{prime|G}}''
are natural isomorphisms then 〈''{{prime|F}}'', ''{{prime|G}}'', {{prime|ε}}, {{prime|η}}〉 is an adjunction where
:<math>\begin{align}
\eta' &= (\tau\ast\sigma)\circ\eta \\
\varepsilon' &= \varepsilon\circ(\sigma^{-1}\ast\tau^{-1}).
\end{align}</math>
Here <math>\circ</math> denotes vertical composition of natural transformations, and <math>\ast</math> denotes horizontal composition.

===Composition===

Adjunctions can be composed in a natural fashion. Specifically, if 〈''F'', ''G'', ''ε'', ''η''〉 is an adjunction between ''C'' and ''D'' and 〈''{{prime|F}}'', ''{{prime|G}}'', ''{{prime|ε}}'', ''{{prime|η}}''〉 is an adjunction between ''D'' and ''E'' then the functor
:<math>F \circ F' : E \rightarrow C</math>
is left adjoint to
:<math>G' \circ G : C \to E.</math>
More precisely, there is an adjunction between ''F F&prime;'' and ''G&prime; G'' with unit and counit given respectively by the compositions:
:<math>\begin{align}
&1_{\mathcal E} \xrightarrow{\eta'} G' F' \xrightarrow{G' \eta F'} G' G F F' \\
&F F' G' G \xrightarrow{F \varepsilon' G} F G \xrightarrow{\varepsilon} 1_{\mathcal C}.
\end{align}</math>
This new adjunction is called the '''composition''' of the two given adjunctions.

Since there is also a natural way to define an identity adjunction between a category ''C'' and itself, one can then form a category whose objects are all [[small category|small categories]] and whose morphisms are adjunctions.

===Limit preservation===

The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore ''is'' a right adjoint) is ''continuous'' (i.e. commutes with [[limit (category theory)|limits]] in the category theoretical sense); every functor that has a right adjoint (and therefore ''is'' a left adjoint) is ''cocontinuous'' (i.e. commutes with [[limit (category theory)|colimits]]).

Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example:
* applying a right adjoint functor to a [[product (category theory)|product]] of objects yields the product of the images;
* applying a left adjoint functor to a [[coproduct]] of objects yields the coproduct of the images;
* every right adjoint functor between two abelian categories is [[left exact functor|left exact]];
* every left adjoint functor between two abelian categories is [[right exact functor|right exact]].

===Additivity===

If ''C'' and ''D'' are [[preadditive categories]] and ''F'' : ''D'' → ''C'' is an [[additive functor]] with a right adjoint ''G'' : ''C'' → ''D'', then ''G'' is also an additive functor and the hom-set bijections

:<math>\Phi_{Y,X} : \mathrm{hom}_{\mathcal C}(FY,X) \cong \mathrm{hom}_{\mathcal D}(Y,GX)</math>
are, in fact, isomorphisms of abelian groups. Dually, if ''G'' is additive with a left adjoint ''F'', then ''F'' is also additive.

Moreover, if both ''C'' and ''D'' are [[additive categories]] (i.e. preadditive categories with all finite [[biproduct]]s), then any pair of adjoint functors between them are automatically additive.