Editing Representable functor (section)

==Relation to universal morphisms and adjoints==

The categorical notions of [[universal morphism]]s and [[adjoint functor]]s can both be expressed using representable functors.

Let ''G'' : ''D'' → ''C'' be a functor and let ''X'' be an object of ''C''. Then (''A'',φ) is a universal morphism from ''X'' to ''G'' [[if and only if]] (''A'',φ) is a representation of the functor Hom<sub>''C''</sub>(''X'',''G''&ndash;) from ''D'' to '''Set'''. It follows that ''G'' has a left-adjoint ''F'' if and only if Hom<sub>''C''</sub>(''X'',''G''&ndash;) is representable for all ''X'' in ''C''. The natural isomorphism Φ<sub>''X''</sub> : Hom<sub>''D''</sub>(''FX'',&ndash;) → Hom<sub>''C''</sub>(''X'',''G''&ndash;) yields the adjointness; that is 
:<math>\Phi_{X,Y}\colon \mathrm{Hom}_{\mathcal D}(FX,Y) \to \mathrm{Hom}_{\mathcal C}(X,GY)</math>
is a bijection for all ''X'' and ''Y''.

The dual statements are also true. Let ''F'' : ''C'' → ''D'' be a functor and let ''Y'' be an object of ''D''. Then (''A'',φ) is a universal morphism from ''F'' to ''Y'' if and only if (''A'',φ) is a representation of the functor Hom<sub>''D''</sub>(''F''&ndash;,''Y'') from ''C'' to '''Set'''. It follows that ''F'' has a right-adjoint ''G'' if and only if Hom<sub>''D''</sub>(''F''&ndash;,''Y'') is representable for all ''Y'' in ''D''.<ref>{{cite book |last=Nourani |first=Cyrus |title=A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos |date=19 April 2016 |publisher=CRC Press |page=28 |isbn=978-1482231502}}</ref>