Early church existed in Rome

CHURCH FATHERS: Epistle to the Romans (St. Ignatius)

In the greetings this explains the Church of Rome

CHURCH FATHERS: Against Heresies, III.3 (St. Irenaeus)

[we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faithpreached to men, which comes down to our time by means of the successions of the bishops.

Succession of bishops (pope)

CHURCH FATHERS: Letter 15 (Jerome)

My words are spoken to the successor of the fisherman, to the disciple of the cross. As I follow no leader save Christ, so I communicate with none but your blessedness, that is with the chair of Peter. For this, I know, is the rock on which the church is built! Matthew 16:18 This is the house where alone the paschal lamb can be rightly eaten. Exodus 12:22 This is the Ark of Noah, and he who is not found in it shall perish when the flood prevails.

Peter is the leader of the church