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This is a modification of the question I previously asked here.

Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we have two pullback squares where all the morphisms are serre fibrations $$\begin{array}& &a&\to &b\\ &\downarrow & &\downarrow \\ &c &\to &d\end{array}$$ $$\begin{array}& &e&\to &f\\ &\downarrow & &\downarrow \\ &g &\to &h\end{array}$$ and maps $f_{ae}:a \to e,f_{bf}:b \to f, f_{cg}:c \to g,f_{dh}:d \to h$ satisfying the obvious commutativity conditions. Then I want to show that if $f_{bf}, f_{cg},f_{dh}$ are weak homotopy equivalence then so is $f_{ae}$.

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