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Such homotopies are given by the $\smile_i$-products. Steenrod gives explicit formulas, IIRC, in [Steenrod, N. E. Products of cocycles and extensions of mappings. Ann. of Math. (2) 48, (1947). 290--320. MR0022071], but the easiest is to prove they exists exist using acyclic models.

(Maybe Steenrod only deals with $\mathbb Z_2$ coefficients? I don't have access to the paper now :( )

Such homotopies are given by the $\smile_i$-products. Steenrod gives explicit formulas, IIRC, in [Steenrod, N. E. Products of cocycles and extensions of mappings. Ann. of Math. (2) 48, (1947). 290--320. MR0022071], but the easiest is to prove they exists using acyclic models.
Such homotopies are given by the $\smile_i$-products.