Is the following a good/accurate definition for an enumerator (a kind of Turing machine)?

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I am working a bit with a book from Sipser (Introduction to the theory of computation, 3rd Ed.).

On page 187 in exercise 3.4 one is asked to give a "..formal definition of an enumerator.[..]"

Is the following ok/correct?

Let $A$ be a recursive enumerable language over alphabet $\Sigma$.

(That would be, as I understood, $A \subset \Sigma^*$, and there is a TM $M_A$ such that for all $a \in \Sigma^*$:

$a\in A $ if and only if $M_A$ accepts $a$.)

Let $E_A$ be a TM with 2 tapes. Then:

We call $E_A$ an enumerator of the language $A$ if and only if

for all $a \in \Sigma^*$: ($a \in A$ if and only if $a$ will eventually appear on the second tape.)

Thank you