Is this the correct understanding of iterative bootstrapping (double precision)?

Is it correct that if I have, for example, two ciphertexts ( c_1 ) and ( c_2 ) and I want to multiply them together, I should apply the iterative bootstrapping to each ( c_1 ) and ( c_2 ) separately and then multiply them afterwards? I urgently need help. Thank you very much.

No, if you have at least one extra level left, you can multiply c_1 by c_2 and then bootstrap the result.