Thank you for watching.I am reading "Algebraic topology from a homotopical viewpoint". At p.106 of this book, there is the following exercise.
4.3.19 Exercise. Let $p:E\rightarrow B$ be a homotopically trivial Hurewicz fibration; i.e.,there exists a homotopy equivalence $\phi: E\rightarrow B\times F$ such that the triangle $$\begin{array}{rcccl} E & \: & \rightarrow & \; & B\times F \\ \:&\searrow & \; & \swarrow &\; \\ \; & \; & B& \; &\; \end{array}$$ commutes, where $\pi: B\times F\rightarrow B$ is the projection, and assume that $(B, A)$ has the HEP. Prove that the induced fibration $p_A:E_A=p^{-1}A\rightarrow A$ is also homotopically trivial.
Let $\psi$ be the homotopy inverse of $\phi$. Since the above diagram commutes,image of the restriction $\phi|_{E_A}$ is in $E_A$.So I think the $\phi |_{E_A}$ is a homotopy equivalence such that $\pi \circ \phi|_{E_A}=p_A$. But I can't find a homotopy inverse of $\phi _{E_A}$.