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Let $A$ and $B$ be simplicial abelian groups, and let $N_\ast(-)$ denote the normalized chain complex functor. Let $$AW_{A,B}\colon N_\ast(A\otimes B) \longrightarrow N_\ast(A)\otimes N_\ast(B)$$ and $$ EZ_{A,B}\colon N_\ast(A)\otimes N_\ast(B) \longrightarrow N_\ast(A\otimes B)$$ denote the Alexander-Whitney map and the Eilenberg-Zilber map respectively. Does anyone know of an explicit chain homotopy realizing $$EZ_{A,B}\circ AW_{A,B}\sim Id_{N_\ast(A\otimes B)}.$$

Motivation for its existence can be found in the comments of this question.

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You have it in page 7 of this paper.

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