Alexander-Whitney map

Definition
The Alexander-Whitney map is a natural transformation between the following two bifunctors on topological spaces: $$(X,Y) \mapsto Sing_.(X \times Y)$$ and $$Sing_.(X) \otimes Sing_.(Y)$$ ($$Sing_.$$ here denotes the singular chain complex functor).

The Eilenberg-Zilber map is a natural transformation in the reverse direction, and the composite both ways is naturally chain-homotopic to the identity transformations on the two functors.

Explicitly the Alexander-Whitney map is given as follows: