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The explicit form of the theorem due to Eilenberg and Mac Lane is in the paper you are referring to "On the groups $H(\Pi,n): I$". A stronger and more precise form is in

Tonks, A.P. On the Eilenberg-Zilber theorem for crossed complexes. J. Pure Appl. Algebra 179 (2003) 199-220.

Also Tonks verifies the important fact that the tensor product is a strong deformation retract of the crossed complex of the product, as is well known for the chain complex case and used in Homological Perturbation Theory. You may find his references to the chain complex case helpful.

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The explicit form of the theorem due to Eilenberg and Mac Lane is in the paper you are referring to "On the groups $H(\Pi,n): I$". A stronger and more precise form is in

Tonks, A.P. On the Eilenberg-Zilber theorem for crossed complexes. J. Pure Appl. Algebra 179 (2003) 199-220.

Also Tonks verifies the important fact that the tensor product is a strong deformation retract of the crossed complex of the product, as is well known for the chain complex case and used in Homological Perturbation Theory. You may find his references to the chain complex case helpful.