proof explanation – Reverse function is enabled.

Continue on this question:

As you can see on this link

$ Phi = f ^ {- 1} circ h $ is well defined and maps $[c,d]$ sure $[a,b]$ now in my textbook the next theorem following what i posted in the link, is the following:

Theorem 2
Let $ f:[a,b] subseteq R to R ^ n $ and $ h:[c,d] subseteq R to R ^ n $ to be 1-1 parametrizations of a simple arc C $ R ^ n $ Then there is a unique function $ Phi $ of [a,b] sure [c,d] such as $ h = f circ Phi $. In addition, $ Phi $ is continuous and strictly monotonous.

Now, I am confused because the proof provided for Theorem 2 is the one provided in the link ("Complementary Question") but in the supplementary question. $ Phi $ maps [c,d] sure [a,b] and not in front. What do I miss?