Right whiskering, unknown naturality

Question

Suppose $F,G:C\to D$ and $H:D\to E$ are functors and $\alpha:F\to G$ is a natural transformation.Let $H\circ \alpha:H\circ F\to H\circ G$ be the right whiskering $(H\circ \alpha)_A:H(FA)\to H(GA)$ defined by $(H\circ\alpha)_A=H(\alpha_A)$.

Now how it follows by naturality of $\alpha$ and functoriality of $H$ that $(H\circ\alpha)_A$ is indeed a natural transformation: $$H(\alpha_B)\circ H(F f)=H(Gf)\circ H(\alpha_A)$$ ?

Answer 1

Directly.
Functors preserve composition, hence also commutative squares.

By naturality of $\alpha$, we already know $\alpha_B\circ Ff=Gf\circ\alpha_A$.
Then just apply $H$ on both sides.