Editing Natural transformation (section)

===Abelianization===
Given a group <math>G</math>, we can define its [[abelianization]] <math>G^{\text{ab}} = G/</math> [[Commutator subgroup#Definition|<math>[G,G]</math>]].  Let 
<math>\pi_G: G \to G^{\text{ab}}</math> denote the projection map onto the cosets of <math>[G,G]</math>. This homomorphism is "natural in 
<math>G</math>", i.e., it defines a natural transformation,  which we now check.  Let <math>H</math> be a group.  For any homomorphism <math>f : G \to H</math>, we have that 
<math>[G,G]</math> is contained in the kernel of <math>\pi_H \circ f</math>, because any homomorphism into an abelian group kills the commutator subgroup.  Then 
<math>\pi_H \circ f</math> factors through <math>G^{\text{ab}}</math> as <math>f^{\text{ab}} \circ \pi_G = \pi_H \circ f</math> for the unique homomorphism 
<math>f^{\text{ab}} : G^{\text{ab}} \to H^{\text{ab}}</math>.  This makes <math>{\text{ab}} : \textbf{Grp} \to \textbf{Grp}</math> a functor and 
<math>\pi</math> a natural transformation, but not a natural isomorphism, from the identity functor to <math>\text{ab}</math>.