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Honestly, those two points don't seem incompatible to me. For example:

Teaching the history of fashion to undergrads in 1985 is bizarre because:

1. They insist that standards of dress are entirely relative. Being dressed decently is a cultural construct; some cultures wear hardly any clothing whatsoever and being nude is a completely normal, default way of presenting yourself.
2. And yet when I walk into class with my dick and balls hanging out, they all get extremely offended and the coeds threaten to call the police.

(And yes I changed the year because I'm sick of so many of these issues being brought up as though "the kids these days" are the problem, when so often these are issues that have been around LITERALLY FOREVER.)

I'm not trying to dunk on this Henry Shelvin guy -- I'm certain that he knows a lot more about philosophy than me, and has more interesting thoughts about morals than I do. And I'm also not going to judge someone based on a tweet...aside from the obvious judgement that they are using Twitter, lol. But as far as takes go, this one kinda sucks.

*edit: I'll add that I hope this professor is taking this opportunity to explain what the difference is between morals being relative vs being subjective, which is an issue that has come up in this very thread. Especially since I bet a lot of his students have only heard the term "moral relativism" being used by religious conservatives who accuse you of being a moral relativist because you don't live by the Bible. I know that was definitely the case for me.