See the definition of the Alexander-Whitney transformation:

http://ncatlab.org/nlab/show/Alexander-Whitney+map

and the Eilenberg-Zilber transformation:

http://ncatlab.org/nlab/show/Eilenberg-Zilber+map

Is there a natural or obvious way to extend them to higher tensor powers i.e to, lets say

$$ \Delta_{A_1,\ldots,A_n} : C(\otimes_{j=1}^n A_j) \to \otimes_{j=1}^n C(A_j) $$ and $$ \nabla_{A_1,\ldots,A_n} : \otimes_{j=1}^n C(A_j) \to C(\otimes_{j=1}^nA_j) $$

or to the infinite tensor power series, such that the adjointness is still there?

(My first obvious guess is to simply iterate them using associativity of the usual tensor product but I'm not sure if it is that simple due to concerns about braiding and singns)