Strictification gives equivalence, not on-the-nose identification

Mac Lane strictification proves that every monoidal category is equivalent to a strict one, not that it is already strict after a harmless renaming of objects. This is exactly the distinction highlighted by Isbell's warning that isomorphic objects cannot simply be identified: forcing specified isomorphisms to become identities can change the category.

The same boundary also separates strictness from skeletality. A skeletal category is one in which isomorphic objects are equal; a skeleton of a given category is a full skeletal subcategory obtained by choosing one representative from each isomorphism class. Strictification instead builds an equivalent monoidal category functorially from the original one. Passing to a skeleton and strictifying are different operations, and they need not be achievable simultaneously.