I'm currently reading about the construction of Steenrod squares in Mosher & Tangora, which starts with defining the cup-$i$ product. In the excerpt below, $W$ is a chain complex with $$W_i = \mathbb{Z}d_i \oplus \mathbb{Z}Td_i,\qquad i\geq 0$$ and differential given by
$$\partial d_i = d_{i-1}+(-1)^i Td_{i-1},\quad \partial T=T\partial,\quad TT=1.$$
(One can think of this as giving a cell structure for $S^\infty$ with two cells in each dimension.)
I'm maybe confused about what $C(\sigma\times\sigma)$ should be. It seems in order for the subsequent discussion to be meaningful, it should have things in degree $>\dim \sigma$. I think the subcomplex in $C(X)\otimes C(X)$ should be $$\mathrm{Span}\big\{\tau\otimes \widetilde{\tau}: \tau,\widetilde{\tau}\text{ are faces of }\sigma\big\},$$ but this doesn't seem to be the image of a subcomplex of $C(X\times X)$ under $\Psi$. If I understand correctly, $\Psi$ is the map which in degree $n$ is given by \begin{align*}\Psi_n\colon C_n(X)\otimes C_n(X)&\to \bigoplus_{i+j=n}C_i(X)\otimes C_j(X)\\ \sigma \otimes \tau &\mapsto \sum_{i+j=n}\sigma|[v_0,\ldots,v_i]\otimes \tau|[v_i,\ldots,v_n].\end{align*} Any help would be greatly appreciated.