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The notion of canonicity (with respect to maps and objects) has thusfar evaded attempts by mathematicians to formalize it. If I remember correctly, Bourbaki tried to give it a definition based on some ideas of Chevalley, but, at least to my knowledge, it was deleted from later drafts of the Elements because it was not an extremely useful notion. Jim Dolan more recently tried to give a definition of a canonical transformation between functors, but his notion is essentially that of a transformation that is natural when restricted to the core groupoid. However, this doesn't really capture all of the cases that we want, and I don't know of any serious attempt to make use of the notion.