> **Puzzle 142.** Prove the associative law: for any natural transformations \\(\alpha: F \Rightarrow G\\), \\(\beta: G \Rightarrow H\\) and \\(\gamma: H \Rightarrow I \\) we have \\((\gamma \beta) \alpha = \gamma (\beta \alpha)\\).

In order to prove that the two composite natural transformations are equal, we show that all their components are equal, that is \\(((\gamma\beta)\alpha)_x = (\gamma(\beta\alpha))_x\\) for all objects \\(x \in \mathcal{C}\\):

\\[
((\gamma\beta)\alpha)_x =
(\gamma\beta)_x\circ\alpha_x =
(\gamma_x\circ\beta_x)\circ\alpha_x =
\gamma_x\circ(\beta_x\circ\alpha_x) =
\gamma_x\circ(\beta\alpha)_x =
(\gamma(\beta\alpha))_x.
\\]

The proof uses the definition of the composite natural transformation and the fact that morphism composition is associative.