Editing Natural transformation (section)

===Tensor-hom adjunction===
{{further|Tensor-hom adjunction|Adjoint functors}}
Consider the [[category of abelian groups|category <math>\textbf{Ab}</math>]] of abelian groups and group homomorphisms. For all abelian groups <math>X</math>, <math>Y</math> and <math>Z</math> we have a group isomorphism
: <math>\text{Hom}(X \otimes Y, Z) \to \text{Hom}(X, \text{Hom}(Y, Z))</math>.
These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors <math>\textbf{Ab}^{\text{op}} \times \textbf{Ab}^{\text{op}} \times \textbf{Ab} \to \textbf{Ab}</math>.
(Here "op" is the [[opposite category]] of <math>\textbf{Ab}</math>, not to be confused with the trivial [[opposite group]] functor on <math>\textbf{Ab}</math> !)

This is formally the [[tensor-hom adjunction]], and is an archetypal example of a pair of [[adjoint functors]]. Natural transformations arise frequently in conjunction with adjoint functors, and indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the ''unit'' and ''counit''.