Editing Representable functor (section)

== Universal elements ==
According to [[Yoneda's lemma]], natural transformations from Hom(''A'',&ndash;) to ''F'' are in one-to-one correspondence with the elements of ''F''(''A''). Given a natural transformation Φ : Hom(''A'',&ndash;) → ''F'' the corresponding element ''u'' ∈ ''F''(''A'') is given by
:<math>u = \Phi_A(\mathrm{id}_A).\,</math>
Conversely, given any element ''u'' ∈ ''F''(''A'') we may define a natural transformation Φ : Hom(''A'',&ndash;) → ''F'' via
:<math>\Phi_X(f) = (Ff)(u)\,</math>
where ''f'' is an element of Hom(''A'',''X''). In order to get a representation of ''F'' we want to know when the natural transformation induced by ''u'' is an isomorphism. This leads to the following definition:
:A '''universal element''' of a functor ''F'' : '''C''' &rarr; '''Set''' is a pair (''A'',''u'') consisting of an object ''A'' of '''C''' and an element ''u'' &isin; ''F''(''A'') such that for every pair (''X'',''v'') consisting of an object ''X'' of '''C''' and an element ''v'' &isin; ''F''(''X'') there exists a unique morphism ''f'' : ''A'' &rarr; ''X'' such that (''Ff'')(''u'') = ''v''.
A universal element may be viewed as a [[universal morphism]] from the one-point set {•} to the functor ''F'' or as an [[initial object]] in the [[category of elements]] of ''F''.

The natural transformation induced by an element ''u'' ∈ ''F''(''A'') is an isomorphism if and only if (''A'',''u'') is a universal element of ''F''. We therefore conclude that representations of ''F'' are in one-to-one correspondence with universal elements of ''F''. For this reason, it is common to refer to universal elements (''A'',''u'') as representations.