In addition, in those days, the primacy of Peter was often address as, “Roman See,” Chair of St. Peter…
Furthermore the ECF writes,
St. Cyprian
In the middle of the third century St. Cyprian expressly terms the Roman See the Chair of St. Peter, saying that Cornelius has succeeded to “the place of Fabian which is the place of Peter” (Ep 55:8; cf. 59:14).
Firmilian of Caesarea
Firmilian of Caesarea notices that Stephen claimed to decide the controversy regarding rebaptism on the ground that he held the succession from Peter (Cyprian, Ep. 75:17). He does not deny the claim: yet certainly, had he been able, he would have done so. Thus in 250 the Roman episcopate of Peter was admitted by those best able to know the truth, not merely at Rome but in the churches of Africa and of Asia Minor.
Tertullian
In the first quarter of the century (about 220) Tertullian (De Pud. 21) mentions Callistus’s claim that Peter’s power to forgive sins had descended in a special manner to him. Had the Roman Church been merely founded by Peter and not reckoned him as its first bishop, there could have been no ground for such a contention. Tertullian, like Firmilian, had every motive to deny the claim. Moreover, he had himself resided at Rome, and would have been well aware if the idea of a Roman episcopate of Peter had been, as is contended by its opponents, a novelty dating from the first years of the third century, supplanting the older tradition according to which Peter and Paul were co-founders, and Linus first bishop.
Hippolytus
About the same period, Hippolytus (for Lightfoot is surely right in holding him to be the author of the first part of the “Liberian Catalogue” – “Clement of Rome”, 1:259) reckons Peter in the list of Roman bishops.
“Adversus Marcionem”
We have moreover a poem, “Adversus Marcionem”, written apparently at the same period, in which Peter is said to have passed on to Linus “the chair on which he himself had sat” (P.L., II 1077).
St. Irenaeus
These witnesses bring us to the beginning of the third century. In the second century we cannot look for much evidence. With the exception of Ignatius, Polycarp, and Clement of Alexandria, all the writers whose works we possess are apologists against either Jews or pagans. In works of such a character there was no reason to refer to such a matter as Peter’s Roman episcopate.
Irenaeus, however, supplies us with a cogent argument. In two passages (Adv. haer. 1:27:1, and 3:4:3) he speaks of Hyginus as ninth Bishop of Rome, thus employing an enumeration which involves the inclusion of Peter as first bishop (Lightfoot was undoubtedly wrong in supposing that there was any doubt as to the correctness of the reading in the first of these passages. In 3:4:3, the Latin version, it is true, gives “octavus”; but the Greek text as cited by Eusebius reads enatos.
Irenaeus we know visited Rome in 177. At this date, scarcely more than a century after the death of St. Peter, he may well have come in contact with men whose fathers had themselves spoken to the Apostle. The tradition thus supported must be regarded as beyond all legitimate doubt.
Lightfoot’s suggestion (Clement 1:64), that it had its origin in the Clementine romance, has proved singularly unfortunate. For it is now recognized that this work belongs not to the second, but to the fourth century. Nor is there the slightest ground for the assertion that the language of Irenaeus, 3:3:3, implies that Peter and Paul enjoyed a divided episcopate at Rome – an arrangement utterly unknown to the Church at any period. He does, it is true, speak of the two Apostles as together handing on the episcopate to Linus. But this expression is explained by the purpose of his argument, which is to vindicate against the Gnostics the validity of the doctrine taught in the Roman Church. Hence he is naturally led to lay stress on the fact that that Church inherited the teaching of both the great Apostles. Epiphanius (“Haer.” 27:6) would indeed seem to suggest the divided episcopate; but he has apparently merely misunderstood the words of Irenaeus.
Those who succeed Peter in Rome succeed him also in the supreme headship
History bears complete testimony that from the very earliest times the Roman See has ever claimed the supreme headship, and that that headship has been freely acknowledged by the universal Church. We shall here confine ourselves to the consideration of the evidence afforded by the first three centuries.
St. Clement
The first witness is St. Clement, a disciple of the Apostles, who, after Linus and Anacletus, succeeded St. Peter as the fourth in the list of popes. In his “Epistle to the Corinthians”, written in 95 or 96, he bids them receive back the bishops whom a turbulent faction among them had expelled. “If any man”, he says, “should be disobedient unto the words spoken by God through us, let them understand that they will entangle themselves in no slight transgression and danger” (Ep. 59). Moreover, he bids them “render obedience unto the things written by us through the Holy Spirit”. The tone of authority which inspires the latter appears so clearly that Lightfoot did not hesitate to speak of it as “the first step towards papal domination” (Clement 1:70). Thus, at the very commencement of church history, before the last survivor of the Apostles had passed away, we find a Bishop of Rome, himself a disciple of St. Peter, intervening in the affairs of another Church and claiming to settle the matter by a decision spoken under the influence of the Holy Spirit. Such a fact admits of one explanation alone. It is that in the days when the Apostolic teaching was yet fresh in men’s minds the universal Church recognized in the Bishop of Rome the office of supreme head.
St. Ignatius of Antioch
A few years later (about 107) St. Ignatius of Antioch, in the opening of his letter to the Roman Church, refers to its presiding over all other Churches. He addresses it as "presiding over the brotherhood of love [prokathemene tes agapes] The expression, as Funk rightly notes, is grammatically incompatible with the translation advocated by some non-Catholic writers, “pre-eminent in works of love”.