In this lecture we prove that the cup product is graded commutative whenever the coefficient ring is commutative. The method of proof is somewhat tortuous, as we first define a second product, the cross product. This product depends on a choice of Eilenberg-Zilber morphism on the cochain level, but is independent of the choice on cohomology.

Next lecture we will show that for a special Alexander-Whitney choice of Eilenberg-Zilber morphism, the cross product (almost) agrees with the cup product. This fact, combined with the uniqueness of diagonal approximations, shows that the cup product is indeed graded commutative as claimed.