Composing functors with natural transformations

Given functors $$\mathcal{A} \xrightarrow{F} \mathcal{B} \overset{G}{\underset{H}{\rightrightarrows}} \mathcal{C} \xrightarrow{K} \mathcal{D}$$ and a natural transformation $\alpha : G \to H$, we can define $\alpha_F$ (also written $\alpha F$) and $K\alpha$ as follows:

• $\alpha_F : G \circ F \to H \circ F$ is the natural transformation whose components are given by $(\alpha_F)_A = \alpha_{FA}$ for $A \in \mathcal{A}$;
• $K\alpha : K \circ G \to K \circ H$ is the natural transformation whose components are given by $(K\alpha)_B = K(\alpha_B)$ for $B \in \mathcal{B}$.

So here, for example, $F\eta : F \to F \circ G \circ F$ has components $(F\eta)_B = F(\eta_B) : FA \to FGFB$.

In my opinion, it becomes clearer if you take the homotopical point of view.

Denote by $\mathcal I$ the interval category: its objects are 0 and 1, and it has a unique non-identity morphism $0\to 1$. Then a natural transformation between $F,G \colon \mathcal C \to \mathcal D$ is just a functor $\alpha \colon \mathcal I \times \mathcal C \to \mathcal D$ making the following commute:

where the vertical arrows are the obvious inclusion at $0$ and $1$.

Then for a functor $H \colon \mathcal D \to \mathcal E$, the functor $H\alpha$ is defined and fit into the diagram:

Similarly, if $J \colon \mathcal B \to \mathcal C$ is a functor, $\alpha J$ is an abuse for the induced functor $\mathcal I \times \mathcal B \to \mathcal D$ fitting in the diagram:

$F\eta$ isn't a composition of a functor and a natural transformation, it's just a natural transformation. If you have a transformation $\alpha:F\to G$, then a little examination will show that there's an induced natural transformation $HF\to HG$ for any $H$ that composes appropriately, often denoted $H\alpha$; and similarly one $FH'\to GH'$ for appropriate $H'$ denoted $\alpha H'$. The component of the former at an object $c$ of $F,G$'s domain is $H(\alpha_c)$; in the latter case, the component at an object $d$ in $H'$'s domain is $\alpha_{H'(d)}$.

Another way to think of it is it's just a bit of an abuse of notation for a horizontal composition of two natural transformations: "$F\eta$" is the horizontal composite $id_F *\eta$, though this may not clarify anything depending on what resources you've learned from.