Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of the proof; I'm specifically trying to get a handle on the methods being used. Please assume I'm asking in the most naive sense possible.
In this article on inference review, it's explained that Mochizuki's approach differs from standard category theory in refusing to identify isomorphic objects.
Since complete ordered fields are rigid, there is really only one way to do this. But for other categories, there are many choices to be made, and the choices must be made in a compatible way. Sometimes it is best to avoid making such choices, but it is possible to do if desired. After all, a pair of equivalent categories cannot distinguish between themselves using only categorical properties. It can thus be a deep theorem to establish such an equivalence, and highly nonobvious.
This way of thinking is becoming more and more entrenched in certain disciplines of mathematics, especially those where category theory has been used extensively. Algebraic geometry is one such discipline. One can, with care, sometimes work as if isomorphic objects are identical. When Mochizuki insists that the isomorphic objects he describes must be distinguished at all costs, and so labelled to keep them distinct, it feels like prohibiting a boxer the use of his fists.
On the other hand, at the beginning of the Wikipedia page on higher category theory it says:
higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.
So isn't higher category theory precisely studying differences between isomorphic objects? Is Mochizuki talking about higher category theory?