Lets parse some more bullshit to set it straight and ignore.
The whole thing could be refuted by the vulgar saying "forcemeat can not be turned back". This is, BTW, end of the story. Nothing in the universe is reversible except increments and decrements of numbers (and all arithmetics). But numbers are pure abstractions and does not exist outside of human's mind, while the patterns, obviosly, do exist.
Try to reverse a molecule of H2O which self-assembles in certain conditions? Does not work? How about reversing arrows of evolution? Of your previous thought which passed trough your consciousness?
This is an example of the notion of inapplicability. All the reversing of arrows" thing is not applicable to anything whatsoever except numbers and abstract linear transformations.
Isomorphism is stated to be a weaker notion of equality -
And then there are the weaker notions of isomorphism, and even weaker of equivalence. which is OK, because Equality is only applicable to abstract concepts, such as Numbers.
Consider this equations which capture the notion of an isomorphism
f . g = id g . f = id
(-1) are simplest example of such morphisms defined for Numbers, or even more general
pred which are defined for Enumerable, which means an Ordered Sequence.
\x -> (pred . succ) x == (succ . pred) x
No wonder that taking one step in any direction and then taking a step back (exactly the opposite direction, exactly the same magnitude) will return to the same initial position.
The confusion arises from the fact that no two physical objects are the same. The notion of Sameness is defined in terms of occupying the same location, not having the same structure. The notion of having the same structure is the notion of Equivalency.
The conclusion is that
This proves that f and g must be the inverse of each other. Therefore any two initial objects are isomorphic.
So, they are trying to prove that two objects are isomorphic by applying mathematical notions to non-mathematical objects which is a type-error.