the bishop of rome doesnt appear until 400 AD as appointed by the emperor. in the late 8th century when ERE reconquered rome one bishop was brought to constaniople and executed for not waiting for imperial approval of his office(which was later delegated to a govenor).
The papacy is officially established when charlemagne invades italy and the bishop of rome agrees to crown him emperor of the romans. the pope is appointed by the emperor of the HRE until 1059