Thomas Nagel's attack on subjectivism is rooted in his view that objective truth is something that cannot be doubted. There are certain elements rooted in logic and language which cannot be escaped -- such as logical truths, and mathematics.  He describes language not as a contingent process, but as logic itself, or a system of concepts to which any language-user must conform. He also claims that language is not thought, but rather is an essential tool of it, and consequentially, logic is a linguistic tool we use to describe our thoughts and engage critically ourselves and others.

As Nagel suggests, when you offer a criticism of someone, what you have offered is an external and a seemingly non-universal point of view.  For this reason, we are inclined to believe that this view, your view, constitutes your own opinion of this other person.  From this, one could conclude that some of our moral, political, and cultural convictions are merely subjective.

However, this view of subjectivism breaks down as we take into account the notion of self-awareness.  The pure externality evaporates when we pose the question, “what are we relying on in ourselves to form that view?”  Nagel claims that these “subjective" convictions are rooted in an inescapable objectivity – inescapable precisely because we cannot understand thought from the outside.  The very concept of subjectivity demands an objective framework in which the subject is located and the special perspective is described.  So your "subjective criticism" of the person was in fact based on premises rooted in objective reason.

Nagel does not deny a system of conventions, nor does he deny that grammar and punctuation require conformity to the linguistic community in an objective sense.  What he does claim is that the validity of those thoughts, those inescapable thoughts that language enables us to express, has its basis in those particular conventions and usages. What is meant by criticism is rooted in convention, while the thought that inspired the premises of the critical sentences is inescapably grounded in objective reason.

The last word in the debate must lie in some unqualified thoughts about how things are, and such thoughts cannot be regarded as mere "psychological dispositions."  The subjectivist gives the last word to justifications, which end in language.  Nagel gives the last word to the justifications themselves.  The justifications are involved in the recognitions which subordinate them, and are implicated in the process of creating such justifications.  Thus, Nagel gives the last word to objective reason, since it is required to substantiate any subjective thoughts at all.