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](https://mathoverflow.net/questions/303851/are-the-alexander-whitney-and-eilenberg-zilber-maps-homotopy-inverse-in-arbitrar).



Explicit chain homotopy for the Alexander-Whitney, Eilenberg-Zilber pair
You have it in page 7 of [this paper](https://arxiv.org/pdf/math/0110308.pdf).


