I'm trying to study the proof of Serre spectral sequence in Hatcher's pdf on spectral sequences. On page 530 he says if we have a Hurewicz fibration $\pi:X\rightarrow B$ with $B$ filtered by skeletons $B^p, p=0,1,2,...$ let $X_p=\pi^{-1}(B^p)$. Hatcher says if $\Phi_A:D_a^p\rightarrow B^p$ is a characteristic map for the $p$-cell $e^p_a$ of $B$, so the restriction of $\Phi_a$ to the boudnary sphere $S_a^{p-1}$ is an attaching map for $e_a^p$. Let $\widetilde{D}_a^p=\Phi_a^\ast(X_p)$, the pullback fibration over $D_a^p$ and let $\widetilde{S}_a^{p-1}$ be the part of $\widetilde{D}_a^p$. We then have a map $\widetilde{\Phi}:\sqcup_a(\widetilde{D}_a^p,\widetilde{S}_a^{p-1})\rightarrow (X_p,X_{p-1})$. Since $B^{p-1}$ is a deformation retract of a neighborhood of $N$ in $B^p$, the homotopy lifting property implies that $\pi^{-1}(N)$ of $X_{p-1}$ in $X_p$ deformation retracts to $X_{p-1}$ in the weak sense. Using the excision property of the homology, this implies that $\widetilde{\Phi}$ induces an isomorphism $\oplus_a H_{p+q}(\widetilde{D}_a^p,\widetilde{S}_a^{p-1})\rightarrow H_{p+q}(X_p,X_{p-1})$.
My question is: I get the paragraph until the last sentence. If we use excision on the pair $(X,\pi^{-1}(N))$, we should get $(\cup_a\pi^{-1}(D_a^p),\pi^{-1}(N)\cap\cup_a\pi^{-1}(D_a^p))$. But this is not quite our pullback fibration. Because while the pullback fibration of an inclusion map is just restriction, in this case we do not have an inclusion map.
Can someone clarify this for me? Thanks a lot!