Let $k$ be a field, and let $X$ and $Y$ be CW-complexes of finite type (although the question makes sense for $k$ a ring and for more general chain complexes of finitely generated free abelian groups). The Künneth and Universal Coefficient Theorems for cohomology give $$ H^n(X\times Y;k) \cong \bigoplus_{i+j=n} H^i\big(X;H^j(Y;k)\big)\supseteq\bigoplus_{i+j=n}\operatorname{Ext}\big(H_{i-1}(X);H^j(Y;k)\big). $$ (Here and elsewhere, where coefficients are omitted we mean $\mathbb{Z}$ coefficients.) Meanwhile, the Künneth Theorem for homology gives $$ H_n(X\times Y) \supseteq \bigoplus_{i+j=n}\operatorname{Tor}\big(H_{i-1}(X);H_j(Y)\big). $$ Now, we have a pairing of cohomology and homology $$ H^n(X\times Y;k)\otimes H_n(X\times Y)\to k, $$ and morally it should be true that this pairing restricted to the Ext and Tor terms should be given by the usual pairing of Ext and Tor, followed by the cohomology-homology pairing for $Y$: $$ \operatorname{Ext}\big(H_{i-1}(X);H^j(Y;k)\big)\otimes \operatorname{Tor}\big(H_{i-1}(X);H_j(Y)\big) \to H^j(Y;k)\otimes H_j(Y)\to k. $$
My question is, does this statement appear in the literature somewhere?