The Papacy
http://www.americancatholictruthsociety.com/docs/ecfpapacy.htm
What did the Early Church Fathers Say?
St. Irenaeus
“The blessed apostles [Peter and Paul], having founded and built up the church [of Rome] . . . handed over the office of the episcopate to Linus” (Against Heresies 3:3:3 [A.D. 189]).
Tertullian
“[T]his is the way in which the apostolic churches transmit their lists: like the church of the Smyrneans, which records that Polycarp was placed there by John, like the church of the Romans, where Clement was ordained by Peter” (Demurrer Against the Heretics 32:2 [A.D. 200]).
The Little Labyrinth
“Victor . . . was the thirteenth bishop of Rome from Peter” (The Little Labyrinth [A.D. 211], in Eusebius, Church History 5:28:3).
Cyprian of Carthage
“The Lord says to Peter: ‘I say to you,’ he says, ‘that you are Peter, and upon this rock I will build my Church, and the gates of hell will not overcome it. . . . ’ [Matt. 16:18]. On him [Peter] he builds the Church, and to him he gives the command to feed the sheep [John 21:17], and although he assigns a like power to all the apostles, yet he founded a single chair [cathedra], and he established by his own authority a source and an intrinsic reason for that unity. . . . If someone [today] does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he [should] desert the chair of Peter upon whom the Church was built, can he still be confident that he is in the Church?” (The Unity of the Catholic Church 4; first edition [A.D. 251]).
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"With a false bishop appointed for themselves by heretics, they dare even to set sail and carry
“Paul testifies that Crescens was sent to Gaul [2 Tim. 4:10], but Linus, whom he mentions in the Second Epistle to Timothy [2 Tim. 4:21] as his companion at Rome, was Peter’s successor in the episcopate of the church there, as has already been shown. Clement also, who was appointed third bishop of the church at Rome, was, as Paul testifies, his co-laborer and fellow-soldier [Phil. 4:3]” (Church History 3:4:9–10 [A.D. 312]).
Council of Sardica
“You cannot deny that you are aware that in the city of Rome the episcopal chair was given first to Peter; the chair in which Peter sat, the same who was head—that is why he is also called Cephas [‘Rock’]—of all the apostles; the one chair in which unity is maintained by all” (The Schism of the Donatists 2:2 [A.D. 367]).
St. Jerome
“[Pope] Stephen . . . was the blessed Peter’s twenty-second successor in the See of Rome” (Against the Luciferians 23 [A.D. 383]).
“Clement, of whom the apostle Paul writing to the Philippians says ‘With Clement and others of my fellow-workers whose names are written in the book of life,’ the fourth bishop of Rome after Peter, if indeed the second was Linus and the third Anacletus, although most of the Latins think that Clement was second after the apostle” (Lives of Illustrious Men 15 [A.D. 396]).