This is probably the most difficult part of this presentation... Suppose we have two functors, F,G: X → Y. A natural transformation φ: F → G is defined when for each object x ∈ X there is an arrow φ(x): F(x) → G(x) in Y, and we have the following property:
 for all f: a → b the equality is true:
G(f) ∘ φ(a) = φ(b) ∘ F(f).
F(a) 
F(f) 
F(b) 
→ 
φ(a)↓ 

↓φ(b) 
G(a) 
→ 
G(b) 
G(f) 
That's why it is called 'natural'  it acts consistently with the functors actions on arrows.
