CW-complexes are defined via characteristic maps rather than from attaching maps, so via maps from $\mathbb D^n$ rather than from $\mathbb S^{n-1}$, because we have the propriety that $\mathbb D^n\times\mathbb D^m=\mathbb D^{m+n}$. I want to define products in a synthetic way, only manipulating objetcs up to homotopy so I need to use the spheres. So basically if $X$ and $Y$ are CW-complexes and $Z$ is the product, to attach one new cell to $Z_{p-1}$ means attaching a product of cells $\mathbb S^{n-1}\to X_{n-1}$ and $\mathbb S^{m-1}\to Y_{m-1}$, where $n+m=p$, and this should give a function $\mathbb S^{n+m-1}\to Z_{p-1}$. We actually already have such a map when the CW-complexes $X$ and $Y$ are the spheres $\mathbb S^{n}$ and $\mathbb S^{m}$, which is the Whitehead map. Indeed we have then $Z_{n+m-1}=\mathbb S^{n}\vee\mathbb S^{m}$ and this implies that the function is of type $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$. Now I want this bit to be generalizable to every product of finite CW-complex (it can have just a finite number of cells if nedded).

We want some function $\mathbb S^{n+m-1}\to Z_{p-1}$ and we would like to do the same thing as in the construction of the map $\mathbb S^{n+m-1}\to \mathbb S^{n}\vee\mathbb S^{m}$, so look a the pushout $\mathbb S^{n-1}\leftarrow \mathbb S^{n-1}\times\mathbb S^{m-1}\to \mathbb S^{m-1}$ but then the attaching maps of the cells are not trivial like they were for products of sphere. This looks very closely like the Whitehead product but I guess there is some non trivial things to consider at this point and I feel like I can't find the good way to tackle the problem.

I would like to know if some work about this had already be done, I'd be very glad that you have some synthetic reference about this problem.

Thank you very much