> **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.