information:extraction, structural:reformatting
Horizontal composition

If η: F ⇒ G {\displaystyle \eta:F\Rightarrow G} is a natural transformation between functors F, G: C → D {\displaystyle F,G:C\to D} and ϵ: J ⇒ K {\displaystyle \epsilon:J\Rightarrow K} is a natural transformation between functors J, K: D → E {\displaystyle J,K:D\to E}, then the composition of functors allows a composition of natural transformations ϵ ∗ η: J ∘ F ⇒ K ∘ G {\displaystyle \epsilon *\eta:J\circ F\Rightarrow K\circ G} with components

By using whiskering (see below), we can write

hence

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 X {\displaystyle X} in category C {\displaystyle C}, (i d i d C) X = i d i d C (X) = i d X {\displaystyle (\mathrm {id} _{\mathrm {id} _{C}})_{X}=\mathrm {id} _{\mathrm {id} _{C}(X)}=\mathrm {id} _{X}}.

As identity functors i d C {\displaystyle \mathrm {id} _{C}} and i d D {\displaystyle \mathrm {id} _{D}} are functors, the identity for horizontal composition is also the identity for vertical composition, but not vice versa.