@AdrianClough Oh, I thought that said commutativity was what we were trying to prove, but now that I read Tim's question more closely I see that he was asking something slightly different so I agree with your argument
I'm still looking for an account of classical obstruction theory for simplicial sets. The reference @OmarAntolín-Camarena gave me doesn't actually prove that the obstruction cocycle is actually a cocycle (it's "obvious").
