This is fairly obscure but the problem he highlights can be overcome easily by localising at the saturation of S_f, for D(f)=D(g) if and only if the saturations are equal, and localising at saturation is an isomorphism
It seems though that we would want the computer to be able to do this kind of reasoning itself, and not rely on humans "pre-resolving" all such problems.
I understood the presentation to be on the distinction between equality and isomorphism and when an isomorphism "is an equality". From the slide on homotopy type theory I get the impression that he finds it unsatisfactory to simply consider all isomorphisms as equalities.
So many times it’s necessary to ‘identify’ two more objects which are isomorphic, and the ‘canonical’ is supposed to justify why this doesn’t cause a problem.
The reason it is necessary here is that the mapping D(f) -> A_f is a priori multi valued, and we need it to be single valued.
However it’s fairly easy to make a definition which is trivially single valued , so I don’t think it’s a very instructive example of the phenomenon.
Probably more pertinent is an n-fold tensor product with different bracket ordering