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'') ∘ ''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 ''G''′, then ''G'' and ''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 ''G''′ then ''F'' is also left adjoint to ''G''′. More generally, if 〈''F'', ''G'', ε, η〉 is an adjunction (with counit–unit (ε,η)) and
:σ : ''F'' → ''F''′
:τ : ''G'' → ''G''′
are natural isomorphisms then 〈''F''′, ''G''′, ε′, 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 〈''F''′, ''G''′, ''ε''′, ''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.