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What would have happened if popes were given the opportunity to appoint their successors? The papacy can't be hereditary because the pope is celibate, but adoption is a possibility here (and Caesar adopted Octavian after all).
We would assume that the cardinals had to meet at some point to confirm the candidate (though whether by a majority or 2/3 supermajority is up for debate).
A conclave would only be called if a pope died without nominating a successor (or if the successor were no longer alive at the time the pope abdicated or died)