quality:correction, information:extraction, linguistic:translation, quality:corruption
Other early Christian mentions

Eusebius (c. 260/265 – 339) mentions Linus as Saint Peter's successor and Clement as the third bishop of Rome in his book Church History. As recorded by Eusebius, Clement worked with Saint Paul as his "co-laborer".

As to the rest of his followers, Paul testifies that Crescens was sent to Gaul; but Linus, whom he mentions in the Second Epistle to Timothy 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.

Tertullian (c. 155 – c. 220 AD) wrote in his work " The Prescription Against Heretics " about the authority of the church in Rome. In this work, Tertullian said that the Church in Rome has the authority of the Apostles because of its apostolic foundation.

Since, moreover, you are close upon Italy, you have Rome, from which there comes even into our own hands the very authority (of apostles themselves). How happy is its church, on which apostles poured forth all their doctrine along with their blood! Where Peter endures a passion like his Lord's! Where Paul wins his crown in a death like John's where the Apostle John was first plunged, unhurt, into boiling oil, and thence remitted to his island-exile!

According to the same book, Clement of Rome was ordained by Saint Peter as the bishop of Rome.

For this is the manner in which the apostolic churches transmit their registers: as the church of Smyrna, which records that Polycarp was placed therein by John; as also the church of Rome, which makes Clement to have been ordained in like manner by Peter.

Optatus the bishop of Milevis in Numidia (today's Algeria) and a contemporary of the Donatist schism, presents a detailed analysis of the origins, beliefs, and practices of the Donatists, as well as the events and debates surrounding the schism, in his book The Schism of the Donatists (367 A.D). In the book, Optatus wrote about the position of the bishop of Rome in maintaining the unity of the Church.

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 —of all the apostles; the one chair in which unity is maintained by all.