Commenting on the development of a one-man episcopacy in Rome in the middle of the second century, Catholic scholar John P. Meier wrote:
In the second half of the second century, the clear affirmation of a monepiscopate in Rome as well as its connection with the apostles is supplied, interestingly, not by Rome itself but by church writers from elsewhere who visit Rome. The exact import of the statement of Hegesippus (c. A.D. 160) concerning episcopal succession in Rome, cited by Eusebius (Ecclesiastical History, 4.22.3), is disputed; but Hegesippus seems at least to attest to the existence of the monepiscopate in Rome circa A.D. 160. Much clearer and explicit evidence is offered by Irenaeus, a theologian of the East who became bishop of Lyons. Somewhere around A.D. 180, Irenaeus visited Rome when Eleutherus was bishop. Irenaeus testifies to the existence of a succession-list of bishops, reaching back to Linus, the first leader of the Roman church supposedly appointed by the apostles Peter and Paul, who are said to have founded the church at Rome (Greek text in Eusebius, E.H. 5.5.8-5.6.1-5; Latin text in Adversus Haereses, 3.3.3.). Tellingly, Peter and Paul are not counted among Rome’s bishops. Indeed, the title “bishop” (episkopos) is absent from the list proper; the abstract noun episkopē is used instead. In the larger context of the list, Irenaeus seems to use interchangeably the titles “bishops” and “presbyter”; elsewhere he speaks of the succession of presbyters in the churches.
Hence, while the names of the Roman leaders in Irenaeus’s list may well record historical figures, the idea that they actually functioned as single bishops in Rome throughout the second century is anachronistic—anachronism being a common failing not only of ancient theologians like Irenaeus but also of ancient historians like Josephus and Tacitus. What Irenaeus’s testimony shows, alongside that of Hegesippus, is that church writers from outside of Rome knew of the monepiscopate existing in Rome from the second half of the second century onwards. Its existence in Rome is not reliably attested before that time. With the convergence of the Simon of history, the Peter of faith, the prominent leadership in Rome, we can begin to speak of the emergence of the papacy in Rome toward the end of the second century. (John P. Meier, “Petrine Ministry in the New Testament and in the Early Patristic Traditions” in John F. Puglisi, ed. How Can the Petrine Ministry Be a Service to the Unity of the Universal Church? [Grand Rapids, Mich.: Eerdmans, 2010], 13-33, here, pp. 29-30)