Editing Natural transformation (section)

===Finite calculus===
For every abelian group <math>G</math>, the set <math>\text{Hom}_\textbf{Set}(\mathbb{Z}, U(G))</math> of functions from the integers to the underlying set of <math>G</math> 
forms an abelian group <math>V_{\mathbb{Z}}(G)</math> under pointwise addition. (Here <math>U</math> is the standard [[forgetful functor]] <math>U:\textbf{Ab} \to \textbf{Set}</math>.) 
Given an <math>\textbf{Ab}</math> morphism <math>\varphi: G \to G' </math>, the map <math>V_\mathbb{Z}(\varphi): V_\mathbb{Z}(G) \to V_\mathbb{Z}(G')</math> given by left composing <math>\varphi</math> with the elements of the former is itself a homomorphism of abelian groups; in this way we 
obtain a functor <math>V_{\mathbb{Z}}: \textbf{Ab} \to \textbf{Ab}</math>. The finite difference operator <math>\Delta_G</math> taking each function 
<math>f: \mathbb{Z} \to U(G)</math> to <math>\Delta(f): n \mapsto f(n+ 1) - f(n)</math> is a map from <math>V_{\mathbb{Z}}(G)</math> to itself, and the collection <math>\Delta</math> of such maps gives a natural transformation <math>\Delta: V_\mathbb{Z} \to V_\mathbb{Z}</math>.