Saint and Sinner wrote:
"Dulles is giving us only the evidence that supports his side. He never mentions Cyprian's two confrontations with Rome's authority or the fact that a later Council of Carthage under Augustine stated that Rome would have no authority to restore a defrocked presbyter to his former position."
It's also worth noting that, in one of his disputes with Rome, Cyprian was joined by dozens of other bishops, including the Eastern bishop (not just North African bishops) Firmilian, who referred to the bishop of Rome as an antichrist and worse than all heretics. That Roman bishop, Stephen, seems to be the first extant advocate of something close to the doctrine of the papacy, but earlier disputes involving the Roman church suggest that previous Roman bishops didn't view themselves as Popes. First Clement is written in the name of the church of Rome, not the bishop of Rome, and the letter makes many appeals to various authorities (scripture, Jesus, the apostles, the Holy Spirit, etc.), but never to any papal authority. Early references to the apostolic nature of the Roman church associate that church with Paul and Peter, without distinguishing between the two by placing Peter over Paul. When Polycrates, the bishop of Ephesus, wrote against the Roman bishop Victor during the Quartodeciman controversy in the second century, he cited two apostles who traveled in the East in support of his position. He may have been citing those two Eastern apostles to counter a Roman appeal to Peter and Paul. It seems that the singling out of Peter was a later development, even in Rome. In the earliest disputes involving the Roman church, there are no traces of a discussion of papal authority on either side.
For those interested in reading more about whether there was a papacy in the earliest generations of Christianity, see here and here.
"they who are at Rome do not observe those things in all cases which are handed down from the beginning, and vainly pretend the authority of the apostles...But with respect to the refutation of custom which they [the Roman church] seem to oppose to the truth, who is so foolish as to prefer custom to truth, or when he sees the light, not to forsake the darkness?...And this indeed you Africans are able to say against Stephen [bishop of Rome], that when you knew the truth you forsook the error of custom. But we join custom to truth, and to the Romans' custom we oppose custom, but the custom of truth; holding from the beginning that which was delivered by Christ and the apostles....But indeed you [Stephen] are worse than all heretics....Moreover, how great sin have you heaped up for yourself, when you cut yourself off from so many flocks! For it is yourself that you have cut off. Do not deceive yourself, since he is really the schismatic who has made himself an apostate from the communion of ecclesiastical unity. For while you think that all may be excommunicated by you, you have excommunicated yourself alone from all...But as far as he [Stephen] is concerned, let us leave him...And yet Stephen is not ashamed to afford patronage to such in opposition to the Church, and for the sake of maintaining heretics to divide the brotherhood and in addition, to call Cyprian 'a false Christ and a false apostle, and a deceitful worker.' And he, conscious that all these characters are in himself, has been in advance of you, by falsely objecting to another those things which he himself ought deservedly to hear." (Firmilian, Cyprian's Letter 74:6, 74:19, 74:23-24, 74:26)