Up To Isomorphism - A Potential “Error”?

Section 4.5 of the CTMU contains the first main text occurrence of the phrase “Up To Isomorphism”. This phrase was used to describe reality theory’s mandate for matter-information equivalence: They must be identical “up to isomorphism”. It’s a phrase Langan uses a fair bit and an important one to understand. Factually, it means the same thing as “A and B are isomorphic”. Semantically it’s an acknowledgement that A and B are not the same and may live in different mathematical universes - so they are the same “up to isomorphism”. This is important for Langan because of Kant’s distinction between our perception of a thing and the “thing in itself”. Langan isn’t throwing out the distinction (at least not yet), but rather disputing the notion that we cannot understand whatever lies behind our perception in any way. Interestingly, he may be using this phrase in error. All of the sources I could find said if A and B are identical up to isomorphism that they may be distinct objects but are isomorphic. In footnote one of section 4.3 however, Langan writes: “In fact, they are identical up to isomorphism beyond which the mental side, being the more comprehensive of the two, outstrips the material.”. The latter part of the sentence suggests to me that the mental side and the material side cannot be isomorphic and what Langan is meaning is that the material side is identical up to isomorphism with some sub-model of the mental side. Another way of saying this is the material side of reality is isomorphically embedded in the mental side. If this is an error, it’s not a serious one and is really more just terminology. I think there is somewhere Langan explains what he means by the phrase “up to isomorphism”, but I don’t remember where.
Anyway, to understand this phrase we will dig in to the concept of isomorphism in the context of model theory, one of the main fields the CTMU draws from. To do this we will learn: 1) What a language is. 2) What a model is. 3) What an isomorphism between models is. 4) What it means for a model to be a sub model of another model and what it means for a model to be isomorphically embedded in another.