natural transformation as functor from the product with the interval category
1. Proposition
Let 
be categories, 
be functors.
Then a natural transformation 
is precisely defined by a functor
making following diagram commutes
see: product category
2. Proof
2.1. natural transformation 
functor
We define
Note that by naturality 
, hence the mapping is welldefined
Then by construction, 
factor through 
now about functorality of 
:
2.1.1. identity
follows from
2.1.2. composition
For composition within 
and 
, this follows from functorality of
For 
we get
where applying functorality of 
results in
For $(f, 1) ˆ (g, 0 → 1) = (f ˆ g, 0 → 1) $ we get
where applying functorality of 
results in
2.2. functor natural transformation
We define
Thus we get a diagram
Note that the Right half commutes, since the first half commutes.
Furthermore, by assumption of factorization, we get 
therefore this construction satisfies naturality