I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem. Link to the statement of the theorem

Basically we want to prove that the Morse homology is independent of the pseudo-gradient field and the morse function. Since the proof is a little bit long and I do not want to just summarize it, here is a link to the proof presented in the book.

http://www.math.stonybrook.edu/~sunscorch/quals/Major/Morse_Invariance.pdf

So, my confusion starts when the author proves that $(C_*(\tilde{F}|_{V \times A}),\partial _{\tilde{X}}) = (C_{*+1}(f_0), \partial_{X_0})$ (page 2 of pdf almost at the end of the page) and similarly for the other equality. After that he states that there are only two trajectories connecting critical points of $\tilde{F}$. So, I have three questions:

- The first type of trajectories are the ones staying always in $A$, and I argued that because of $(C_*(\tilde{F}|_{V \times A}),\partial _{\tilde{X}}) = (C_{*+1}(f_0), \partial_{X_0})$. So, like analyzing the trajectories in such section is just as analyzing the trajectories of $X_0$, similarly for $X_1$. Is this the correct idea?
- I do not get why there can only be trajectories from critical points of $f_0$ to those of $f_1$ (which are the second type of trajectories between critical points of $\tilde{F}$). My first thought about why we cannot have trajectories from critical points of $f_1$ to $f_0$ is because of how their indexes are related to the indexes of $\tilde{F}$ (almost at the beginning of page 2 of pdf) , but I am not sure why this would be true. Any suggestion?
- My final question is in the definition of $\partial_{\tilde{X}}$. I do not know why in the first row, second column of its matrix representation, it is the zero matrix. Any ideas? (My hypothesis is that since we can only have trajectories as those described in the second bullet point, the number of trajectories from critical points of $f_1$ to those of $f_0$ and by definition of $\partial_{\tilde{X}}$, then that is why it is the zero matrix).

Again, I would really appreciate your help here. Maybe this is the easiest part of the proof and I am missing a basic fact, but I do not see it. Thanks in advance.