Editing Adjoint functors (section)

===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]]