Homotopic maps give fiber homotopy equivalent fibrations

Recall the mapping-path fibration \[ E_f=\{(a,\gamma):\gamma(0)=f(a)\}, \qquad p_f(a,\gamma)=\gamma(1). \] Let \(H:A\times I\to B\) be a homotopy from \(f\) to \(g\), and write \(H_a(t)=H(a,t)\). Define \[ \Phi:E_f\to E_g, \qquad \Phi(a,\gamma)=(a,\overline{H_a}*\gamma), \] where \(\overline{H_a}\) is the reverse path from \(g(a)\) to \(f(a)\). This is fiber-preserving because the endpoint of the concatenated path is the endpoint of \(\gamma\).

Using \(H_a\) instead gives \[ \Psi:E_g\to E_f. \] The composites prepend the backtracking paths \[ H_a*\overline{H_a} \quad\text{or}\quad \overline{H_a}*H_a. \] Contract these backtracking paths to the constant path by the standard reparametrization homotopy. This produces fiber-preserving homotopies \[ \Psi\Phi\simeq1_{E_f}, \qquad \Phi\Psi\simeq1_{E_g}. \] Hence \[ \boxed{E_f\to B\text{ and }E_g\to B\text{ are fiber homotopy equivalent}.} \]