§1. When the episcopal see is vacant, the coadjutor bishop immediately becomes the bishop of the diocese for which he had been appointed provided that he has legitimately taken possession of it.
§2. When the episcopal see is vacant and unless competent authority has established otherwise, an auxiliary bishop preserves all and only those powers and faculties which he possessed as vicar general or episcopal vicar while the see was filled until a new bishop has taken possession of the see. If he has not been designated to the function of diocesan administrator, he is to exercise this same power, conferred by law, under the authority of the diocesan administrator who presides over the governance of the diocese.
Source: Code of Canon Law (Vatican.va)