The usual way to compose two natural transformation is what is known as the vertical composition. Given categories $\mathcal{C},\mathcal{D}$ , functors $R,S,T$ from $\mathcal{C}$ to $\mathcal{D}$ , and natural transformations $\tau:R\Rightarrow S$ and $\eta:S\Rightarrow T$ , we have a natural transformation $\eta\bullet \tau:R\Rightarrow T$ given by $$(\eta \bullet \tau)_A:=\eta_A\circ \tau_A$$ The reason for calling $\bullet$ the ``vertical'' composition is illustrated by the diagram below:

However, there is another way to compose natural transformations: the so-called horizontal composition. Given categories, $\mathcal{B},\mathcal{C},\mathcal{D}$ , functors $S_1,T_1: \mathcal{B}\to \mathcal{C}$ , $S_2,T_2:\mathcal{C}\to \mathcal{D}$ , and natural transformations, $\tau: S_1\Rightarrow T_1$ and $\eta: S_2 \Rightarrow T_2$ as in the following diagram

we define the horizontal composition, or Godemont product, of $\eta$ and $\tau$ , written $\eta\circ \tau: S_2S_1\to T_2T_1$ , as follows: first, pick any object $A$ in $\mathcal{B}$ . Because $\eta$ is a natural transformation, we have a commutative diagram (solid arrows) below

Below are some properties of $\circ$ :

1. $\eta\circ \tau$ is a natural transformation.
2. $\circ$ is associative.
3. $\eta\circ 1_S=\eta$ , and $1_S\circ \tau =\tau$ , where $\eta$ and $\tau$ are described in the diagram above, and $1_S$ is the identity transformation on the functor $S:\mathcal{B}\to \mathcal{C}$ , and $1_T$ is the identity transformation on the functor $T:\mathcal{C}\to \mathcal{D}$ .
4. $\circ$ and $\bullet$ satisfy the interchange law.

In fact, the first three properties above turn ${Cat}$ , the category of small categories, into a category where the objects are small categories, morphisms are natural transformations, and the composition of morphisms is the horizontal composition.