Corinth was not a church down the street from Rome, it was a city in far away Greece, outside of Clement's local jurisdiction, yet within his universal jurisdiction at the same time.
St Clement wasn’t ordained directly from Christ, none of the successors are. We have the Apostles who acts a Bishops ordaining Bishops and Deacons and their successor’s keeps this continuity as an unbroken chain of succession. Tertullian is also very clear that since Peter was Bishop of Rome, St Clement had to be ordained in the likeness of St Peter.
“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”
[Prescription Against Heretics 32 (c. A.D. 200)].
So Clement was not only Bishop of Rome like St Peter was, but also a Pope.
Eusebius of Cesarea:
Paul testifies that Crescens was sent to Gaul [2 Tm 4: 10], but Linus, whom he mentions in the Second Letter to Timothy [2 Tm 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] [ibid., 3: 4: 9–10].
St Jerome: 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
[Illustrious Men 15 (A.D. 392)].
St Clement of Rome had authority:
Owing, dear brethren, to the sudden and successive calamitous events that have happened to us, we feel that we have been somewhat tardy in turning our attention to the points about which you consulted us; and especially to that shameful and detestable sedition, utterly abhorrent to the elect of God, that a few rash and self-confident persons have kindled to such a pitch of frenzy that your venerable and illustrious name, worthy to be universally loved, has suffered grievous injury. . . . It is right and holy, therefore, men and brethren, to obey God rather than to follow those who, through pride and sedition, have become the leaders of a detestable emulation. For we shall incur no slight injury, but rather great danger, if we rashly yield ourselves to the inclinations of men who aim at exciting strife and tumults, to draw us away from what is good . . . we may reach the goal set before us in truth wholly free from blame. Joy and gladness you will afford us, if you become obedient to the words written by us and through the Holy Spirit root out the lawless wrath of your jealousy according to the intercession we have made for peace and unity in this letter...
[Letter to the Corinthians I, 14, 63 (A.D. 70)].
Dionysius and the continuing importance of Clement’s letter:
“In the same letter he [Dionysius] refers to Clement’s Epistle to the Corinthians, proving that from the very first it had been customary to read it in church. He says: ‘Today being the Lord’s Day, we kept it as a holy day and read your epistle, which we shall read frequently for its valuable advice, like the earlier epistle which Clement wrote on your behalf.’”
With much evidence that St Clement was a successor of St Peter and recognizes his authority as others did, the historical writings proved to be irrefutable.