Consider a finite dimensional CW complex $X$ and let $C^*(X)$ denote the cochain complex of its cellular cochains. We have the diagonal map $\Delta: X \rightarrow X^2$ sending $x \mapsto (x,x)$. This induces a map:
$C^*(X)\otimes C^*(X)\rightarrow C^*(X^2)\rightarrow C^*(X)$
Which we call multiplication of cellular cochains. The first arrow above is the isomorphism of tensor and cross product and the second arrow is the map induced by the cellular approximation of the diagonal.
It seems to be common knowledge that this turns $C^*(X)$ into a DG algebra I.e. it satisfies the differential liebnitz rule. I was wondering wether anyone knows how this is proved or could provide a reference.