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This is a follow-up to those two questions: Cartesian product of two CW-complexes, and Product of CW complexes question.

Consider two cell complexes $A$ (with cells $e^m_\alpha$ and attaching maps $\varphi_\alpha$) and $B$ (with cells $e^n_\beta$ and attaching maps $\varphi_\beta$). Then $A \times B$ has a structure of a cell complex, with cells the products $e^m_\alpha \times e^n_\beta$ and attaching maps $\varphi_\alpha \times \varphi_\beta$. When applying the boundary operator to $D^m_\alpha \times D^n_\beta$, one obtains $(\partial D^m_\alpha \times D^n_\beta) \cup (D^m_\alpha \times \partial D^n_\beta)$.

My question is: how are the attaching maps $\varphi_\alpha \times \varphi_\beta$ defined with regard to the individual $\varphi_\alpha$ and $\varphi_\beta$ maps ? Apparently, the product of cell complexes is unique, so how are the attaching maps uniquely described ? How are the elements of $(\partial D^m_\alpha \times D^n_\beta) \cup (D^m_\alpha \times \partial D^n_\beta)$ mapped to the $(m+n-1)$ skeleton of the product cell complex ?

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A particular element $\partial D^m_\alpha \times D^n_\beta$ of $\partial(D^m_\alpha \times D^n_\beta)$, is mapped to $\varphi_\alpha(\partial D^m_\alpha) \times D^n_\beta$ which is part of the $(m + n − 1)$ skeleton.

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