Studying Algebraic Topology from Tom Dieck I came across cup product. I was trying to verify whether there were a way to prove associativity without using the direct formula of cup, just exploit the naturality of cap product thanks to its definition on chain level (depending on Eilenberg-Zilber Theorem for example using the Alexander-Whitney map).
In particular in my notes there's the following diagram, but I don't know what should suggest me
$$\begin{array}{ccccccccc} X & \overset{\triangle}{\rightarrow} & X \times X & \overset{1 \times \triangle}{\rightarrow} X \times(X \times X) \\\ & & \downarrow{\triangle \times 1} & \\\ & & (X \times X) \times X \end{array}$$
Where $\triangle$ is the diagonal map and $1$ is the identity.
Any help or answer would be appreciated.