Editing Natural transformation (section)

==Definition==
If <math> F</math> and <math> G</math> are [[functor]]s between the categories <math> C</math> and <math> D </math> (both from <math> C</math> to <math> D</math>), then a '''natural transformation''' <math> \eta</math> from <math> F</math> to <math> G</math> is a family of morphisms that satisfies two requirements.
# The natural transformation must associate, to every object <math> X</math> in <math> C</math>, a [[morphism]] <math>\eta_X :  F(X) \to G(X)</math> between objects of <math> D </math>. The morphism <math> \eta_X</math> is called the '''component''' of <math> \eta</math> at <math> X </math>.
# Components must be such that for every morphism <math> f :X  \to  Y</math> in <math> C</math> we have:

:::<math>\eta_Y \circ F(f) = G(f) \circ \eta_X</math>

The last equation can conveniently be expressed by the [[commutative diagram]]

[[File:Natural Transformation between two functors.svg|center|This is the commutative diagram which is part of the definition of a natural transformation between two functors.]]

If both <math> F</math> and <math> G</math> are [[contravariant functor|contravariant]], the vertical arrows in the right diagram are reversed. If <math> \eta</math> is a natural transformation from <math> F</math> to <math> G </math>, we also write <math> \eta : F \to G</math> or <math> \eta : F \Rightarrow G </math>. This is also expressed by saying the family of morphisms <math> \eta_X: F(X) \to G(X)</math> is '''natural''' in <math> X </math>.

If, for every object <math> X</math> in <math> C  </math>, the morphism <math> \eta_X</math> is an [[isomorphism]] in <math> D </math>, then <math> \eta</math> is said to be a '''{{visible anchor|natural isomorphism}}''' (or sometimes '''natural equivalence''' or '''isomorphism of functors'''). Two functors <math> F </math> and <math> G</math> are called ''naturally isomorphic'' or simply ''isomorphic'' if there exists a natural isomorphism from <math> F</math> to <math> G</math>.

An '''infranatural transformation''' <math> \eta</math> from <math> F</math> to <math> G</math> is simply a family of morphisms <math>\eta_X : F(X) \to G(X) </math>, for all <math> X</math> in <math> C</math>. Thus a natural transformation is an infranatural transformation for which <math> \eta_Y \circ F(f)  = G(f) \circ \eta_X</math> for every morphism <math> f : X \to Y </math>.  The '''naturalizer''' of <math> \eta </math>, nat<math> (\eta) </math>, is the largest [[subcategory]] of <math> C</math> containing all the objects of <math> C</math> on which <math> \eta</math> restricts to a natural transformation.