Proof Yoneda lemma

this diagram shows natural transformation Φ determined




Φ

a


(


i
d


a


)
=
u


{\displaystyle \phi _{a}(\mathrm {id} _{a})=u}

since each morphism f : → x 1 has

Φ

x


(
f
)
=
(
f
f
)
u
.


{\displaystyle \phi _{x}(f)=(ff)u.}

moreover, element u∈f(a) defines natural transformation in way. proof in contravariant case analogous.