>> “They are claiming Clement was a disciple of Peter” <<
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Where do we find any evidence of Peter having disciples?
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“Eubulus greets you, also Pudens and Linus and Claudia and all the brethren.” - 2 Tim 4:21
Linus was Peter’s successor, the second Pope. He was a disciple of both Paul and Peter. He was martyred. I’ve heard that Clement, the fourth Pope, was a disciple of John. By disciple, it is not meant that the apostles took the place of Jesus. It just means that they were close followers of this apostle or that apostle.