linguistic:translation, quality:correction, structural:summarization
Interchange law

The two operations are related by an identity which exchanges vertical composition with horizontal composition: if we have four natural transformations α, α ′, β, β ′ {\displaystyle \alpha,\alpha ',\beta,\beta '} as shown on the image to the right, then the following identity holds:

Vertical and horizontal compositions are also linked through identity natural transformations:

As whiskering is horizontal composition with an identity, the interchange law gives immediately the compact formulas of horizontal composition of η: F ⇒ G {\displaystyle \eta:F\Rightarrow G} and ϵ: J ⇒ K {\displaystyle \epsilon:J\Rightarrow K} without having to analyze components and the commutative diagram: