I believe this is germane to this topic - the teaching of Pope Saint Gregory the Great that the three Patriarchates which
existed in his time -Rome, Alexandria and Antioch- founded by Peter, were equal in power and authority. This is something from
one of the Church's greatest Popes which you will never
see in any Catholic work on the papacy.
1. The parts where the Pope speaks of Alexandria and Antioch sharing the keys with Rome
2. The parts where the Pope speaks of the equality of Rome and Alexandria and Antioch
3. The parts where the Pope says that all three of these Sees form one See over which the three bishops preside.
St Gregory I, Pope of Rome, Epistle XL, writing to Pope Eulogius
Patriarch of Alexandria.
"Your most sweet Holiness [Eulogius] has spoken much in your letter to me about
the chair of Saint Peter, Prince of the apostles, saying that he
himself now sits on it in the persons of his successors.
"And indeed I acknowledge myself to be unworthy, not only in the
dignity of such as preside [he means the bishops], but even in the
number of such as stand [he means the faithful].
But I gladly accepted all that has been said, in that he has spoken to
me about Peter's chair who occupies Peter's chair. …And to him it is
said by the voice of the Truth, To thee I will give the keys of the
kingdom of heaven (Matth. xvi. 19). And again it is said to him, And
when thou art converted, strengthen thy brethren (xxii. 32). And once
more, Simon, son of Jonas, lovest thou Me? Feed my sheep (Joh. xxi.
Wherefore though there are many apostles, yet with regard to the
principality itself the See of the Prince of the apostles alone has
grown strong in authority, which in three places is the See of one.
For he himself [Peter] exalted the See in which he deigned even to
rest and end the present life [Rome
]. He himself adorned the See to
which he sent his disciple as evangelist [Alexandria
]. He himself
established the See in which, though he was to leave it, he sat for
seven years [Antioch
]. Since then it is the See of one, and one See,
over which by Divine authority three bishops now preside, whatever
good I hear of you, this I impute to myself.”
(Book VII, Epistle XL)