Fiber-homotopy equivalence

In algebraic topology, a fiber-homotopy equivalence is a map over a space "B" that has homotopy inverse over "B" (that is we require a homotopy be a map over "B" for each time "t".) It is a relative analog of a homotopy equivalence between spaces.
Given maps "p":"D"→"B", "q":"E"→"B", if ƒ:"D"→"E" is a fiber-homotopy equivalence, then for any "b" in "B" the restriction
is a homotopy equivalence. If "p", "q" are fibrations, this is always the case for homotopy equivalences by the next proposition.
Proof of the proposition.
The following proof is based on the proof of Proposition in Ch. 6, § 5 of . We write formula_2 for a homotopy over "B".
We first note that it is enough to show that ƒ admits a left homotopy inverse over "B". Indeed, if formula_3 with "g" over "B", then "g" is in particular a homotopy equivalence. Thus, "g" also admits a left homotopy inverse "h" over "B" and then formally we have formula_4; that is, formula_5.
Now, since ƒ is a homotopy equivalence, it has a homotopy inverse "g". Since formula_6, we have: formula_7. Since "p" is a fibration, the homotopy formula_8 lifts to a homotopy from "g" to, say, "g'" that satisfies formula_9. Thus, we can assume "g" is over "B". Then it suffices to show "g"ƒ, which is now over "B", has a left homotopy inverse over "B" since that would imply that ƒ has such a left inverse.
Therefore, the proof reduces to the situation where ƒ:"D"→"D" is over "B" via "p" and formula_10. Let formula_11 be a homotopy from ƒ to formula_12. Then, since formula_13 and since "p" is a fibration, the homotopy formula_14 lifts to a homotopy formula_15; explicitly, we have formula_16. Note also formula_17 is over "B".
We show formula_17 is a left homotopy inverse of ƒ over "B". Let formula_19 be the homotopy given as the composition of homotopies formula_20. Then we can find a homotopy "K" from the homotopy "pJ" to the constant homotopy formula_21. Since "p" is a fibration, we can lift "K" to, say, "L". We can finish by going around the edge corresponding to "J":