The Primacy by the other churches appear at the end of the 1st century…St. Ignatius elevated the Roman community over all the communities using in his epistle a solemn form of address. Twice he says of it that it is the presiding community, which expresses a relationship of superiority and inferiority.
The First Council of Constantinople (AD 381) suggested strongly that Roman primacy was already asserted. However, it should be noted that, because of the controversy of this claim, the Pope did not personally attend this ecumencial council that was held in the capital of the eastern empire, rather than at Rome. It was not until 440 that Leo the Great more clearly articulated the extension of papal authority as doctrine, promulgating in edicts and in councils his right to exert “the full range of apostolic powers that Jesus had first bestowed on the apostle Peter”.
It was at the Ecumenical Council of Chalcedon in 451 that Leo I (through his emissaries) stated that he was “speaking with the voice of Peter”.
At this same Council, an attempt at “compromise” was made when the bishop of Constantinople was given a primacy of honour only second to that of the Bishop of Rome, because “Constantinople is the New Rome.” Ironically, Roman papal authorities rejected this language since it did not clearly recognize Rome’s claim to juridical authority over the other churches.
The Catholic Church doctrine of the “sedes apostolica” (apostolic see) of Leo I which states that every bishop of Rome, as Peter’s successor, possesses the full authority granted to this position. This power, then, is inviolable on the grounds that it was established by God himself and so not bound to any individual. Leo I (440-461), with the aid of Roman law, solidified this doctrine by making the bishop of Rome the legal heir of Peter. According to Leo, the apostle Peter continued to speak to the Christian community through his successors as bishop of Rome.
Second Council of Lyons
On 29 June (Feast of Peter & Paul patronal feast of Popes), Gregory X celebrated a Mass in St John’s Church, where both sides took part. The council declared that the Roman church possessed “the supreme and full primacy and authority over the universal Catholic Church.” {Wikipedia}
Where isn’t the Church of Rome considered "the supreme and full primacy and authority over the universal Catholic Church.”?
St. Irenaeus of Lyon’s “Against Heresies” (3:3:2): “With [the Church of Rome], because of its superior origin, all the churches must agree… and it is in her that the faithful everywhere have maintained the apostolic tradition.”
Romes Role as Arbiter
This passage in Irenaeus [from Against Heresies 3:4:1] illuminates the meaning of his remarks about the Church of Rome: if there are disputes in a local church, that church should have recourse to the Roman Church, for there is contained the Tradition which is preserved by all the churches. Rome’s vocation [in the pre-Nicene period] consisted in playing the part of arbiter, settling contentious issues by witnessing to the truth or falsity of whatever doctrine was put before them. Rome was truly the center where all converged if they wanted their doctrine to be accepted by the conscience of the Church. They could not count upon success except on one condition – that the Church of Rome had received their doctrine – and refusal from Rome predetermined the attitude the other churches would adopt. There are numerous cases of this recourse to Rome
en.wikipedia.org/wiki/Historical_development_of_the_doctrine_of_Papal_Primacy