natural transformation from contravariant hom functor as element
1. Proposition
Let 
be a locally small category, 
an object and 
a contravariant functor and 
the contravariant hom-functor.
Then a natural transformation
is naturally given by an element 
, hence
2. Proof
we show, that
2.1. construction
2.1.1. a)
2.1.2. b)
2.2. welldefined
is a 
as 
2.2.2. b)
2.2.2.1. naturality
we get
since by construction, it holds, that
we may write (using sloppy notation) 
Hence it remains to show, that
Since we are working in set, we may element chase, hence for 
where since 
is contravariant, it holds, that
2.3. bijection
2.3.1. a)
Given a natural transformation 
, we get
Here by element chasing for 
and pseudo commutativity of a natural transformation we get
2.3.2. b)
for 
, it holds, that
