Isbell's warning that isomorphic objects cannot simply be identified

Mac Lane records Isbell's objection to the idea that one can make every monoidal category strict merely by identifying isomorphic objects. In a skeleton of $\mathbf{S}\mathbf{e}\mathbf{t}$, choose an infinite set $D$ with $D\cong D\times D$ and note that both projections from $D\times D$ are epic.

If one then tried to force the associator to be the identity in that skeleton, the projection formulas collapse so far that every two endomorphisms $f,g:D\to D$ become equal. That is absurd: the identity of $D$ and any nontrivial constant endomorphism are different. So strictification is a theorem about equivalence, not a license to replace isomorphic objects by equal ones without changing the category.