While studying the book Higher Topos Theory I have encountered some difficulty with Lemma 3.3.4.1, which says that the pullback along a cartesian fibration of a map q such that $q^{op}$ is cofinal is a cartesian equivalence (i.e. a weak equivalence in the model structure on marked simplicial sets) once we declare the marked 1-simplices in both the domain and codomain to be the cartesian edges with respect to the appropriate cartesian fibrations. But this seems to imply that such a q is a weak categorical equivalence, i.e. a weak equivalence in the Joyal model structure, at least when it's between fibrant objects.

This essentially because we can consider the pullback along the identity map on the codomain of q, which surely is a cartesian fibration, and then use the fact that the cartesian model structure on marked simplicial sets is Quillen equivalent to the Joyal one.

Now, this fact is clearly false, since we can pick one of the two maps $\Delta^0 \to \Delta^1$ which is not a categorical equivalence.

Am I missing something or is there a problem in the proof? I think, if ever, the problem might lie in the commutative square that appears in the proof, and in the statement that the upper horizontal arrow is a Joyal equivalence.