> **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.

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.