$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 2.5.8.11. Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. Then the Alexander-Whitney homomorphism
\[ \mathrm{AW}: \mathrm{N}_{\ast }(A \otimes B) \rightarrow \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \]
is a quasi-isomorphism: that is, it induces an isomorphism on homology.
Proof.
By virtue of Proposition 2.5.8.9, the Alexander-Whitney homomorphism is a left inverse to the Eilenberg-Zilber map $\mathrm{EZ}: \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B) \rightarrow \mathrm{N}_{\ast }(A \otimes B)$, which is a quasi-isomorphism by virtue of Theorem 2.5.7.14.
$\square$