While I'm reading the Algebra: Chapter 0 by Paolo Aluffi, I've encountered the following paragraph where he discuss that the isomorphism on categories is too restrictive:
When are two categories ‘equivalent’?
Essentially every notion of ‘isomorphism’ encountered so far has boiled down to ‘structure-preserving bijection’, and the reader may well expect that the natural notion identifying two categories should be drawn from the same model: a functor matching objects of one category precisely with objects of another, preserving the structure of morphisms.
One could certainly introduce such a notion, but it would be exceedingly restrictive. Recall that solutions to universal problems, that is, just about every construction we have run across, are only defined up to isomorphism. Requiring solutions in one context to match exactly with solutions in another would be problematic. The structure of an object in a category is adequately carried by its isomorphism class”, and a natural notion of ‘equivalence’ of categories should aim at matching isomorphism classes, rather than individual objects. The morphisms are a more essential piece of information; the quality of a functor is first of all measured on how it acts on morphisms.
My problem lies in the part
Requiring solutions in one context to match exactly with solutions in another would be problematic.
while I can understand that the notion of functor isomorphism seems too restrictive on categories, I do want to further convince myself by an actual example. Is there an example that can justify the sentence "requiring solutions in one context to match exactly with solutions in another would be problematic"?