Trait by trait doesn’t seem like a likely copy means.
One hemisphere, then the other, almost does though.
I find this idea disturbing because it implies that emulating any brain (and possibly copying de novo AI as well) would inevitably result in creating and destroying multiple different personality/value sets that might count as separate people in some way. No one has ever brought this up as an ethical issue about uploads as far as I know (although I have never read “Age of Em” by Robin Hanson), and my background is not tech or neuroscience, so there is probably something I am missing .
Suppose, as you were waking up, different parts of the brain would ‘come online’. In theory, it could be the same thing. (With the ‘incomplete parts’ running even.)