at.algebraic topology – Induced map in homology for a map to a loop space

Suppose $Y$ is an $(n-1)$-connected space, $n>2$, so we have Hurewicz isomorphisms $pi_n(Y)cong H_n(Y)$ and $pi_{n-1}(Omega Y)cong H_{n-1}(Omega Y)$. Let a map $alphacolon XtoOmega Y$ be given. Naturally it induces a map $betacolon Xtimes S^1to Y$. I want to show the following diagram is commutative:
$$require{AMScd}
begin{CD}
H_{n-1}(X) @>times(S^1)>> H_n(Xtimes S^1)\
@Valpha_*VV @Vbeta_*VV \
H_{n-1}(Omega Y) @<cong<< H_n(Y).
end{CD}
$$

Since $Omega Y$ is path connected, we may homotope $alpha$ to a based map. Then $beta$ factors though the reduced suspension $Sigma X$. If $X=S^{n-1}$ is a sphere, the commutativity would then follow from tracking down the definition of $pi_n(Y)xrightarrow{cong}pi_{n-1}(Omega Y)$. However I don’t know how this helps for the general case.

One can also phrase the question in cohomology in the obvious way. (In particular the cross product $times(S^1)$ will be replaced by the slant product $/(S^1)$.)