Besides the fact that in the very Acta of the Council of Ephesus, the list of names has the following for St. Cyril of Alexandria:
“By Cyril of Alexandria, who managed the place of the most holy and sacred [Pope] Celestine, Archbishop of the Roman Church” (translation from Ryan Grant)
The Greek and Latin can be seen side-by-side at this link.
But besides that, I came across an interesting citation while reading Fr. Francis Dvornik’s “The Idea of Apostolicity in Byzantium” (Page 135). Apparently, when Emperor Anastasius had inquired to Pope Anastasius II (497) on how to resolve the Acacian schism, some legates from the Patriarch of Alexandria had given a letter to the Papal legates in Constantinople. That letter says the following:
“The venerable holy churches of the cities of Rome and Alexandria have always preserved concord, not only in the true and immaculate faith from the time the word of salvation was preached in them, but also in divine ministry. That is to say in both of them the foundation of faith was laid by the same man — we mean the blessed Apostle Peter, whose imitator in everything was the holy evangelist Mark – so that, whenever it happened that in times of uncertainty some councils of bishops were due to be held, the most holy man who presided over the church of Rome used to delegate the most reverend archbishop of the city of Alexandria to take his place” (Collectio Avellana, pg 469) For Latin , see link.
So, obviously this is a lot of pomp, but it shows that in A.D. 497, the Patriarch of Alexandria thought that in the past the Roman bishop had delegated to the Bishop of Alexandria to hold his place at Councils. No doubt, Cyril at Ephesus is in mind.
In the historical context just before the Council of Ephesus, Pope Celestine’ wrote an epistle to Cyril with the instructions for Nestorius’ excommunication, and herein he delegates Cyril with the following words:
“And so, appropriating to yourself the authority of our see, and using our position, you shall with resolute severity carry out this sentence [of excommunication]….” (P.L. 50, 463)
The Council of Ephesus itself made reference to this Papal letter in its own epistle to Emperors Theodosius and Valentinian in July 11th, 431:
“And now they have approved with one accord our sentence concerning the faith, and those who differed they have pronounced to be cut off from the priesthood. And before the assembly of this synod, Celestine, bishop of great Rome, showed by his letter to Cyril, beloved of God, bishop of the great city of Alexandria, whom he appointed to act in his place….” (source)
And as almost all know, the Papal legates at the Council made it clear that the Bishop of Rome was the head of the Council, and that the decisions of the bishops to unite themselves to their Head was most appropriate. I quote here below:
“Philip the presbyter and legate of the Apostolic See said: ‘There is no doubt, and in fact it has been known in all ages, that the holy and most blessed Peter, prince (ἔξαρχος) and head of the Apostles, pillar of the faith, and foundation (θεμέλιος) of the Catholic Church, received the keys of the kingdom from our Lord Jesus Christ, the Saviour and Redeemer of the human race, and that to him was given the power of loosing and binding sins: who down even to today and forever both lives and judges in his successors. The holy and most blessed pope Cœlestine, according to due order, is his successor and holds his place, and us he sent to supply his place in this holy synod, which the most humane and Christian Emperors have commanded to assemble, bearing in mind and continually watching over the Catholic faith. ‘ ” (Session III)
“Philip, presbyter and legate of the Apostolic See said: We offer our thanks to the holy and venerable Synod, that when the writings of our holy and blessed pope had been read to you, the holy members by our [or your] holy voices, you joined yourselves to the holy head also by your holy acclamations. For your blessedness is not ignorant that the head of the whole faith, the head of the Apostles, is blessed Peter the Apostle.” (Session II)
I think there is enough evidence here to suggest that the Council of Ephesus accepted the Roman Primacy at least on Petrine grounds, and that his judgment was revered as coming from the one who presides over Councils. I think, at the very least, this should serve as a corrective to those who would argue that the Eastern christians never accepted the Petrine origin of Rome’s primacy, and that it was Peter who was the rock and keyholder over the Kingdom of Christ. Philip the legate even says the succession of Peter is perpetual, without any contest. But, we would need more information to settle the significance of that.
I think Dom Christopher Butler nicely summarized the initiative of Pope St. Celestine (A.D. 430) when he was notified of the obstinate heresy of the Patriarch of Constantinople, Nestorius, and the latter’s own perspective on the matter:
“This very brief resume shows that the bishop of Rome had been prepared to excommunicate the Bishop of Constantinople from the Catholic Church on his own authority, and that Cyril of Alexandria was ready to act on the Pope’s mandate in executing this sentence; that when the Emperor interposed a Council, the Council regarded itself as obliged to carry out the Pope’s instructions; and that, on a matter which those instructions had not foreseen (the case of John of Antioch and his fellows), the Council’s action was not regarded as final by the Pope, whose desire for a more peaceful solution was accepted and effectuated. Cyril was, of course, the leading personality at the Council, and one can sympathize with the complaint of the condemned Nestorius, who writes:
‘Who was judge ? Cyril. Who was accuser? Cyril. Who was Bishop of Rome? Cyril. Cyril was everything’
Thus Nestorius feels that the authority of the head of the Church was being abused by his own bitterest enemy. On the other hand, of Eutyches, who later fell into the opposite error to his own, Nestorius says: ‘He had received judgment. What other judgment was requisite beyond that which the bishop of Rome had made?’ ” (The Church and Infallibility, Dom B.C. Butler – Page 174)