There is no evidence that his teaching/letter was infallible.
If it was the apostles themselves who eyewitnessed the death and resurrection of Jesus, who were given the divine command to be sent to preach and teach, including writing under the inspiration of the Holy Spirit. The blessed apostles handed these down to Popes Linus, AnaCletus and Clement.
Clement “declares” to the Corinthian Church what was given to them by Sacred Tradition the Faith, when the Pope Clement’s letter calls them to the peace and teachings of the apostles.
This is what is meant by infallibility, that the Pope when speaking ex-Cathedra from the Chair of Peter, just like St.Irenaeus points out that St.Clement writes to the Corinthian Church what the pope declares is infallible because what he reveals to the Corinthian Church is what is already been revealed by Jesus and the Apostles handed down to them. These teachings when taught ex-Cathedra are infallible without error holding all the Church’s to them, which are the traditions and teachings from the Apostles.
St.Irenaeus; "The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric. This man, as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoeing in his ears, and their traditions before his eyes. …In the time of this Clement, no small dissension having occurred among the brethren at Corinth,
the Church in Rome despatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their faith, and declaring the tradition which it had lately received from the apostles.(Against Heresies, Bk. 3, Chap.3)
Because the powerful letter came from the Roman Church, it came from St.Peter’s apostolic successor. It is from this Church that all other Church’s are to follow as per Sacred Tradition.
Correct and infallible and not necessarily mutually inclusive. Additionally, even if it was, there is, as you say, no objection by his colleagues, but agreement.
St.Ignatius of Antioch writes to Pope Clement in regards to his letter to the Corinthian Church, but never disagree’s with Pope Clement’s declaration nor questions the infallible content which came from the apostles by faith and the apostolic traditions.
This declaration Pope Clement makes to the Corinthian Church applies to all the Church’s because it declares the teachings of the Apostles and their traditions handed down to the Church.
This same infallibility is what Vatican I describes, when the Popes speak ex-Cathedra on faith and morals, these are the same teachings from the apostles, but declared in a different age maybe in a different language, never the less this infallible statement will never conflict nor contradict the apostolic teachings and traditions.