Editing Natural transformation (section)

=== Horizontal composition ===

If <math>\eta: F \Rightarrow G</math> is a natural transformation between functors <math>F, G: C \to D</math> and <math>\epsilon: J \Rightarrow K</math> is a natural transformation between functors <math>J, K: D \to E</math>, then the composition of functors allows a composition of natural transformations <math>\epsilon * \eta: J \circ F \Rightarrow K \circ G</math> with components
:<math>(\epsilon * \eta)_X = \epsilon_{G(X)} \circ J(\eta_X) = K(\eta_X) \circ \epsilon_{F(X)}</math>.
By using whiskering (see below), we can write
:<math>(\epsilon * \eta)_X = (\epsilon G)_X \circ (J \eta)_X = (K \eta)_X \circ (\epsilon F)_X</math>,
hence
:<math>\epsilon * \eta = \epsilon G \circ J \eta = K \eta \circ \epsilon F</math>.

[[Image:Horizontal composition of natural transformations.svg|center|400px|alt=This is a commutative diagram generated using LaTeX. The left hand square shows the result of applying J to the commutative diagram for eta:F to G on f:X to Y. The right had side shows the commutative diagram for epsilon:J to K  applied to G(f):G(X) to G(Y).]]

This horizontal composition of natural transformations is also associative with identity.
This identity is the identity natural transformation on the [[identity functor]], i.e., the natural transformation that associate to each object its [[identity morphism]]: for object <math>X</math> in category <math>C</math>, <math>(\mathrm{id}_{\mathrm{id}_C})_X = \mathrm{id}_{\mathrm{id}_C(X)} = \mathrm{id}_X</math>.
:For <math>\eta: F \Rightarrow G</math> with <math>F, G: C \to D</math>, <math>\mathrm{id}_{\mathrm{id}_D} * \eta = \eta = \eta * \mathrm{id}_{\mathrm{id}_C}</math>.
As identity functors <math>\mathrm{id}_C</math> and <math>\mathrm{id}_D</math> are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa.<ref>{{cite web | url=https://bartoszmilewski.com/2015/04/07/natural-transformations/ | title=Natural Transformations | date=7 April 2015 }}</ref>