Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set:

• $X_p = \pi^{-1}(B^p)$,
• $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, the kernel of a map induced by the inclusion $X_p \to X$.

A. Hatcher in his spectral sequences book writes on p.25 that:

The cup product in $H^∗(X; R)$ restricts to maps $F^m_p \times F^n_s \to F^{m+n}_{p+s}$.

The argument for that is that $F^m_p$ can be regarded as the image of the map $H^m(X, X_{p−1}) \to H^m(X)$ via the exact sequence of the pair $(X, X_{p−1})$, and then uses commutativity of a certain diagram (bottom of p. 26 in the mentioned book), part of which is a map $H^{m+n}(X\times X, X_p \times X \cup X \times X_s) \to H^{m+n}(X\times X, (X\times X)_{p+s})$. My question is: what sort of map is this? I I think it is induced by the inclusion $(X\times X)_{p+s} \to X_p \times X \cup X \times X_s$, and I think the proof requires this map to be a monomorphism (otherwise I don't see how to obtain commutativity of the diagram). But how to obtain the last statement?