Yoneda Embedding

Let be a Small Category.
The collection of Covariant Hom-Functors
and Natural Transformations

when
define a Functor:

called the Yoneda embedding.

Dually, the Contravariant Hom-Functors
together with Natural Transformations

when
define a Functor:

which we also call the Yoneda embedding.

Lemma

Yoneda embedding is a Functor.

Proof

Follows from Associativity of composition in .

Lemma

The Yoneda embedding defines a Full and Faithfull Functor

Proof

Putting in Yoneda Lemma,
we find that the Yoneda embedding is a bijection from the set
to the collection of Natural Transformations
Thus this will automatically be Full and Faithfull.