Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for Betti cohomology with $R$-coefficient $$ H_{sing}^*(\alpha): H_{sing}^*(X,R) \to H_{sing}^*(Y,R). $$

My question is that this correspondence satisfies the functoriality condition? That is, take a triple of smooth projective complex varieties $X$, $Y$ and $Z$, let $p$ (resp. $q$, $r$) be the natural projection $X\times Y \times Z \to X \times Y$ (resp. $X\times Y \times Z \to Y\times Z$, $X\times Y \times Z \to X\times Z$), for $\alpha\in CH^*_R(X\times Y)$ and $\beta\in CH^*(Y\times Z)_R$, is there an equality of morphisms $$ H^*_{sing}(r_*(p^*\alpha \cdot q^*\beta)) \overset{?}{=} H^*_{sing}(\beta) \circ H^*_{sing}(\alpha) $$

$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for Betti cohomology with $R$-coefficient $$ H_{\sing}^*(\alpha): H_{\sing}^*(X,R) \to H_{\sing}^*(Y,R). $$

My question is that this correspondence satisfies the functoriality condition? That is, take a triple of smooth projective complex varieties $X$, $Y$ and $Z$, let $p$ (resp. $q$, $r$) be the natural projection $X\times Y \times Z \to X \times Y$ (resp. $X\times Y \times Z \to Y\times Z$, $X\times Y \times Z \to X\times Z$), for $\alpha\in CH^*_R(X\times Y)$ and $\beta\in CH^*(Y\times Z)_R$, is there an equality of morphisms $$ H^*_{\sing}(r_*(p^*\alpha \cdot q^*\beta)) \overset{?}{=} H^*_{\sing}(\beta) \circ H^*_{\sing}(\alpha) $$