Sullivan has half a chapter on Irenaeus. Here’s a little taste:
“According to Irenaeus, Peter and Paul, not Peter alone, appointed Linus as the first in the succession of bishops of Rome. This suggests that Irenaeus did not think of Peter and Paul as bishops, or of Linus and those who followed as successors of Peter more than of Paul.” (From Apostles to Bishops, 148-149.)
About the same time as Irenaeus, Tertullian wrote about the Roman tradition of “apostolic succession.”
“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.” (Tertullian, Prescription Against Heretics 32, in ANF 3:258.) So around 200 AD, the Roman tradition was that Peter ordained the THIRD bishop of Rome–Clement–after Linus and Anicetus. If that’s so, then Peter obviously wasn’t the first bishop of Rome.