'anti-papacy' is a modern compound formed from the prefix 'anti-' and the noun 'papacy'. 'anti-' originates from Greek 'antí' where it meant 'against', and 'papacy' originates from Medieval Latin 'pāpātus' (from 'papa') where 'papa' meant 'father' (a title for the bishop of Rome).

'papacy' passed into English via Medieval Latin 'pāpātus' and Old French 'papauté' (or similar forms) and became the English noun 'papacy'; in Modern English the prefix 'anti-' was attached to form 'anti-papacy' to denote opposition to that institution.

Initially the elements simply meant 'against' + 'the office of the pope/father'; over time the compound came to specifically denote political or religious opposition to the papacy as an institution.