Name convention for functor and natural transformation composition If there are functors $H: D \to C; F,G: C \to D$ and $K: B \to C$ and a natural transformation $\alpha: F \xrightarrow{.} G$ then we can construct 2 new natural transformations:
Aka "left composition"
$$H \alpha : H F \xrightarrow{.} H G $$
and "right composition"
$$ \alpha K : F K \xrightarrow{.} G K $$
I was not able to find the naming convention for these ones and called them as left and right compositions but not sure that there are correct namings. Could anybody help me in the finding correct ones?

It's called whiskering; you can show that it is the same as the horizontal composition of $\alpha$ with $1_H:H\Rightarrow H$ (for your "left composition") or with $1_K:K\Rightarrow K$ (for your "right composition").
