This is enough to show that it is quite wrong to contend that there was no concern for marking out or keeping inviolate the contents of the new covenant Scriptures in the second century, or to claim that there was no generally accepted core canon at least by the end of the third century. In fact, at many points it looks as though the orthodox church at this time was overconfident in its assumption of unanimity.
…With or without the Muratorian Fragment, there is ample evidence that the church was operating with a conception of a closed canon at least by the latter half of the second century. For writers of that generation like Irenaeus and Serapion to speak of NT writings as that which was “passed on” to them from the previous generation (Irenaeus Ag. Her. 3.1.1; Serapion, Hist eccl. 6.12.3–6) shows that they did not conceive the question the Hahneman does. No matter how many new “Gospels” or imitation apostolic epistles continued to appear, there was no chance of any of them making there way into the church’s body of Scriptures. Apart from the occasional rearguard efforts necessitated by controversialists such as Gaius of Rome, the question that rises to face the church by the end of the second century has mostly to do with minor variations that may have existed in the traditional collections held in the archives of individual apostolic churches.
—C. E. Hill, “The Debate Over the Muratorian Fragment and the Development of the Canon,” Westminster Theological Journal 57 (1995), 451–452.