I am a bit confused about how the "product attaching-maps" work.
For instance, if I wanted to find the cell structure for $S^1\times S^1$ then I can proceed similarly to here.
So we will call $e_0$ and $f_0$ the zero-cells of $S^1$ and $S^1$, respectively. Similarly, $e_1$ and $f_1$ are the respective one-cells. Then $\phi$ and $\psi$ are the attaching maps taking the boundary of $D^1$ to $e_0$ and $f_0$ respectively.
When we attach the cell $e_1\times f_1$ to the one-skeleton of $S^1\times S^1$, we do so via the attaching map $\phi\times \psi$. But $\phi\times \psi$ are only defined on $\partial D^1 \times \partial D^1$, and not on $\partial (D^1\times D^1) = (\partial D_1\times D_1) \cup (D_1\times \partial D_1)$. This is where I get confused.
If anyone can explain what's going on I would highly appreciate it.