Peterson's gripes with postmodernism mainly stem from his hatred for Marxism and Neo-Marxism, whatever that means to him. He falsely equates the two by claiming that the Neo-Marxists attempt to the shroud their ideology in the form of "oppressionism" which is a staple of postmodernism. This is obviously wrong for a number of reasons but mainly because; 1) Marxism is rooted in scientific praxis by way of dialectical materialism which postmodernism opposes. 2) Marxism presents clear cut structures and systems for how society operates. Postmodernism says that systems are oppressive in nature and therefore should be opposed.

Now, i mentioned all this not because I'm a Marxist, but because i disagree with the two characteristics of postmodernism i outlined. Firstly, i don't think it is in the best interest of any discipline to disagree with scientific reasoning simply because of how it has proven time and time again to be efficient. Science is the absolute and it should rather be science dismantling other modes of thinking than the other way around. Secondly, I disagree with the proposition that systems are oppressive therefore should be abandoned. Society, politics, etc CAN and SHOULD be explained in clear cut systems once again simply because of how accurate they are. It may not be the conflict theory, but there are other theories out there that explain society and accurately predict the actions of those in it e.g rational choice theory, feminist theory, etc.

I can't claim to have read Foucault or Derrida or any other postmodernist literature, but this is my understanding of some of its claims.

Also, i have thought about Peterson potentially being a postmodernist (or at least incorporating some elements of postmodernist thought into his own ideologies) especially if you watched his recent discussion with Richard Dawkins. The way he tried to reconcile both points of view and say that they are actually two sides of the same coin seemed a bit postmodern to me. Although, what do i know.