Whiskering and horizontal composition of natural transformations

Let $\alpha:F\Rightarrow G:\mathcal C\to\mathcal D$ be a natural transformation (Natural transformation and its components), and let $H:\mathcal D\to\mathcal E$ and $K:\mathcal B\to\mathcal C$ be functors (Covariant functor, identity functor, composite functor, and contravariant functor). The left whiskering $H\alpha:HF\Rightarrow HG$ has components $H(\alpha_A)$, and the right whiskering $\alpha K:FK\Rightarrow GK$ has components $\alpha_{KB}$.

For $\beta:H\Rightarrow L:\mathcal D\to\mathcal E$, the horizontal composite $\beta*\alpha:HF\Rightarrow LG$ has either equal component formula

$(\beta*\alpha)_A=\beta_{GA}\circ H(\alpha_A)=L(\alpha_A)\circ\beta_{FA}.$

Their equality is the naturality equation for $\beta$ at $\alpha_A$. Naturality of the resulting family is proved in Horizontal composites of natural transformations satisfy naturality ↗.