Editing Natural transformation (section)

=== Whiskering ===

Whiskering is an [[external binary operation]] between a functor and a natural transformation.<ref>{{cite web | url=https://proofwiki.org/wiki/Definition:Whiskering | title=Definition:Whiskering - ProofWiki }}</ref><ref>{{cite web | url=https://ncatlab.org/nlab/show/whiskering | title=Whiskering in nLab }}</ref>

If <math>\eta: F \Rightarrow G</math> is a natural transformation between functors <math>F, G: C \to D</math>, and <math>H: D \to E</math> is another functor, then we can form the natural transformation <math>H \eta: H \circ F \Rightarrow H \circ G</math> by defining
:<math>(H \eta)_X = H(\eta_X)</math>.
If on the other hand <math>K: B \to C</math> is a functor, the natural transformation <math>\eta K: F \circ K \Rightarrow G \circ K</math> is defined by
:<math>(\eta K)_X = \eta_{K(X)}</math>.

It's also an horizontal composition where one of the natural transformations is the identity natural transformation:
:<math>H \eta = \mathrm{id}_H * \eta</math> and <math>\eta K = \eta * \mathrm{id}_K</math>.
Note that <math>\mathrm{id}_H</math> (resp. <math>\mathrm{id}_K</math>) is generally not the left (resp. right) identity of horizontal composition <math>*</math> (<math>H \eta \neq \eta</math> and <math>\eta K \neq \eta</math> in general), except if <math>H</math> (resp. <math>K</math>) is the [[identity functor]] of the category <math>D</math> (resp. <math>C</math>).