I am trying to find out whether there are mathematically important or useful properties (of some object(s)) that are nevertheless not invariant under some usual choice of isomorphism?
Are there any such "natural" examples of objects having some property (expressed in some language) which are non-invariant?
What I mean by "natural" is roughly that I want a non ad-hoc example, i.e. something actually used/known by some mathematician(s) that they find important for some purpose or other.
The background to my questioning is that I would like to probe whether some of the claims made by structuralist philosophers of mathematics, specifically that "structural" properties are the only mathematically relevant properties, hold any water. By structural properties I mean properties invariant under isomorphisms in some category (in whatever relevant sense).
So far, I have not been able to find a convincing counter-example (one that isn't ad-hoc) to the above structuralist claim, hence my question.