One of the definitions of NL is the set of all languages that can be decided by a logarithmic space verifier with a 'read-once' witness tape, i.e. the witness tape head can only move right.I want to prove that if we allow the verifier to read the witness tape twice, i.e.  when the witness tape head reaches the end of the tape, it goes back to the beginning and we can read it once more, the set of languages that can be decided by such verifier is still NL.I thought of proving it by saying that, given a 'read-twice' verifier, $V$, which expects a witness $w$, we can build a 'read-once' verifier, $V'$, which works the same as $V$, but expects a witness of the form $ww$, i.e the originial witness concatenated to itself. Is that correct? If not- why not?Thanks!
Equivalence of NL verifier definitions
complexity theory