Proofreading is like quality assurance: at this stage, the typeset first proofs are thoroughly checked for mistakes – both in the text itself and in the typesetting. This stage takes about a month.
The author will carefully read through their first proofs (see typesetting) and flag any mistakes they might see, but Unbound also uses professional proofreaders. A proofreader reads through the first proofs with an eagle eye to make sure no mistakes get through. This could be any remaining spelling or punctuation errors, as well as typesetting errors like a misplaced line break or hyphen. They give the manuscript its final polish.
The author and proofreader both send their corrections back to the in-house editor at Unbound, who collates the two sets of corrections and passes them back to the typesetter. The typesetter then takes in the corrections and generates a revised set of proofs: the second proofs.
At this stage, authors can no longer make changes to the text – except for in extreme cases – as the risk of introducing new errors is too high, and the copy editing and proofreading stages are now behind us!
The second proofs are then checked in-house to make sure all of the corrections have been taken in correctly. This all happens over the course of a month or so. When the in-house editor is satisfied that all corrections have been taken in, the typesetter generates final proofs.
(And here is some info on the publishing process as a whole.)