In Canon 833, the Catholic Church requires various categories of persons to make a formal profession of faith and, by a later rule enacted by John Paul II, some of these are required to make an oath of fidelity to the magisterium. The text of the profession and oath is here.
Among the categories of persons required to make the profession and take the oath are "teachers in any universities whatsoever who teach disciplines pertaining to faith or morals [docentes qui disciplinas ad fidem vel mores pertinentes in quibusvis universitatibus tradunt]." It is striking that this is not restricted to professors in Catholic universities. Nor does this appear to be restricted, in the way that the requirement of the Mandatum is, to those who teach as theologians. For "disciplines pertaining to faith or morals" surely includes, at least, those who teach moral philosophy, and perhaps those who teach philosophy of religion as well.
But what is most striking to me is the title, which I assume is official, for the text of the profession and oath: "Profession of Faith and Oath of Fidelity on Assuming an Office to be Exercised in the Name of the Church". This implies that the Catholic professor may never teach disciplines relating to faith or morals on her own, even if she is at a non-Catholic university. What pertains to the Gospel must always be taught by her in the name of the Body of Christ. This is formalized for professors, but I think is true in everyday life, too. This is scary—but the flip side of this is that if we are speaking in the name of the Body of Christ we can draw on the Church's resources, intellectual and spiritual (and especially sacramental), and trust in the Holy Spirit when teaching, not worrying so much about what we are going to say (cf. Mt. 10:19—not that students are very much like Roman torturers!).