Given spectral sequence of algebras $E_r$ over field $k$, it is clear that there is induced map for $H(E_r)\otimes H(E_r)\to H(E_r\otimes E_r)$ where $H$ is taken against the homology against differential of the page $r$.
$\textbf{Q:}$ Why for all page $r$, $H(E_r)\otimes_kH(E_r)\cong H(E_r\otimes_k E_r)$?
Ref. McCleary, User's guide to spectral sequences, Chpt 1, Def 1.7
In McCleary's setting, $k$ is a field, so this is an application of the Künneth isomorphism.