Editing Natural transformation (section)

=== Interchange law ===

The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations <math>\alpha, \alpha', \beta, \beta'</math> as shown on the image to the right, then the following identity holds:
:<math> (\beta' \circ \alpha') * (\beta \circ \alpha) = (\beta' * \beta) \circ (\alpha' * \alpha)</math>.
Vertical and horizontal compositions are also linked through identity natural transformations:
:for <math>F: C \to D</math> and <math>G: D \to E</math>, <math>\mathrm{id}_G * \mathrm{id}_F = \mathrm{id}_{G \circ F}</math>.<ref>https://arxiv.org/pdf/1612.09375v1.pdf, p. 38</ref>

As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of <math>\eta: F \Rightarrow G</math> and <math>\epsilon: J \Rightarrow K</math> without having to analyze components and the commutative diagram:
:<math>\begin{align}
\epsilon * \eta & = (\epsilon \circ \mathrm{id}_J) * (\mathrm{id}_G \circ \eta) = (\epsilon * \mathrm{id}_G) \circ (\mathrm{id}_J * \eta) = \epsilon G \circ J \eta \\
                & = (\mathrm{id}_K \circ \epsilon) * (\eta \circ \mathrm{id}_F) = (\mathrm{id}_K * \eta) \circ (\epsilon * \mathrm{id}_F) = K \eta \circ \epsilon F
\end{align}</math>.