Show that the single-tape TMs that can not write on the portion the portion of the tape containing the input string recognize only regular languages.

The first part of the answer in a book said that:

We give a DFA $A$ that is equivalent to TM $M$. Design $A$ in two stages. 

1. We arrange the states and transitions of $A$ to simulate $M$ on the read-only input portion of the tape.

2. We assign the accept states of $A$ to correspond to the action of $M$ on the read/write portion of the tapeto the right of the input.

Given below the whole answer (sorry for using pictures but the answer was so long and I do not understand it and I have not enough time to write it):

[![enter image description here][1]][1]
[![enter image description here][2]][2]
[![enter image description here][3]][3]

But I do not understand many points:

1. Why the domain and the codomain of the defined function takes this form? and why only finitely many such functions exist as stated by the last line in the first paragraph?

2. Why the author wrote $F_{\epsilon}$ as in the third paragraph? did he used the idea of epsilon transitions?

3. The last paragraph in the solution I did not understand its idea at all?

Could anyone clarify the above questions for me please?

I saw an answer for the question here https://cs.stackexchange.com/questions/22082/single-tape-turing-machines-with-write-protected-input-recognize-only-regular-la but I did not understand it at all.

    


  [1]: https://i.sstatic.net/ONnvn.png
  [2]: https://i.sstatic.net/kLqqq.png
  [3]: https://i.sstatic.net/tjOXb.png