Let $F_1, G_1: C \rightarrow D, \text{ and } F_2, G_2: D \rightarrow E$ be two functors of categories, and let $\alpha: F_1 \Rightarrow G_1$, $\beta: F_2 \Rightarrow G_2$ be natural transformations of these functors.
Then we can construct a natural transformations $F_2 F_1 \Rightarrow G_2 G_1$ via the following compositions:
$$ F_2 F_1 (x) \xrightarrow{F_2(\alpha_x)} F_2 G_1(x) \xrightarrow{\beta_{G_1(x)}} G_2 G_1(x)$$
$$ F_2 F_1 (x) \xrightarrow{\alpha_{F_1(x)}} G_2 F_1(x) \xrightarrow{G_2(\beta_x)} G_2 G_1(x)$$
Due to the naturality of $\alpha, \beta$ these two compositions are supposed to give the same natural transformation $F_2 F_1 \Rightarrow G_2 G_1$. Why is this the case?