TITLE: Natural isomorphisms: what is the status now of "the Eilenberg/Mac Lane Thesis"?
QUESTION [13 upvotes]: I asked this some days ago over at math.se, and while the question got 10 upvotes, I didn't get too many answers. Although it is a "soft question", maybe the general issue is interesting enough to raise again here in the hope of more answers. 
The Church/Turing Thesis that we all know and love asserts that every algorithmically computable function (in an informally characterized sense) is in fact recursive/Turing computable/lambda computable. A certain intuitive concept, the Thesis claims, in fact picks out the same functions as certain (provably equivalent) sharply defined concepts. 
Evidence? Two sorts: (1) "quasi-empirical", i.e. no unarguable clear exceptions have been found, (2) conceptual, as in for example Turing's own efforts to show that when we reflect on what we mean by algorithmic computation we get down to the sort of operations that a Turing machine can emulate.
OK, now compare. The "Eilenberg/Mac Lane Thesis" in one version (but does anyone call it that??) is that if an isomorphism between widgets and wombats is intuitively "natural" (i.e. doesn't depend on arbitrary choices of co-ordinates, or the like) then it can be regimented as an natural isomorphism between suitable functors in the official category-theoretic sense. A certain intuitive concept, the Thesis claims, in fact picks out the same isomorphisms as a certain sharply defined concept.
Evidence? We'd expect two sorts. (1*) "quasi-empirical", i.e. no clear exceptions. (2*) conceptual ...
Two main questions:
(A) But are there no exceptions? Or (to take the perhaps more likely direction of failure) are there well known cases where we can say "Hey, this is the sort of isomorphism whose intuitive naturalness was surely of the kind that Eilenberg/Mac Lane were trying to chraracterize, back in the day: but actually, you can't shoehorn this case into the framework of their theory of natural isomorphisms."
(B) Assuming the Thesis isn't defeated by counter-example (or is true for some nice class of cases), what are the best efforts at trying to show that, conceptually, it "ought" to be true (when it is)?
And then I suppose that there is a supplementary question arising of category theoretic etiquette:
(C) Assuming again the Thesis isn't obviously defeated by counter-example, is it secure enough for it now to be acceptable to slide without further argument from showing that there is an isomorphism between widgets and wombats constructed in an intuitively "natural", unarbitrary, way to announcing that there is therefore a natural isomorphism here?

REPLY [5 votes]: There is probably no definite answer to this question. But let me propose a few ideas:
1) this kind of general "thesis" only makes sense for naturality with respect to isomorphisms:
There is plenty of examples of perfectly natural and canonical construction (in a naive sense) that are not functorial with respect to arbitrary morphisms:
The center of a group or an algebra, the group of automorphism of some object etc... are example of construction that are functorial only with respect to isomorphisms.
For example of a natural transformation which is only functorial on isomorphims not an isomorphism, consider the category $P$ of partially ordered set and monotone map between them. Consider the functors, $F:P \rightarrow sets$ the functor that forget the ordering, and $B: P \rightarrow Set$ that send any poset to $\{0,1\}$ you can then define a map from $F(p) \rightarrow B(p) = \{0,1\}$ for any posets $p$ as the map which send any element to $1$ if it is a maximal element and $0$ otherwise. It is not functorial with respect to arbitrary morphism but is functorial with respect to isomorphisms.
2) I think there is actually reasonable hopes to be able to give a formal statement and a prof of the isomorphisms part of the thesis: one can imagine purely categorical (and somehow tautological) argument for that of course, or just the philosophical stand that "if a construction does not depends on any choice it cannot distinguishes isomorphic structures", even better I like to take "functorial with respect to isomorphism" as the definition of "canonical".
But what I had in mind here is the Univalence axiom and the developement of category theory with homotopy type theory as it is developed in the Hott book, which I think goes a lot further along those ideas:
To sum things up, in this formalism every category is 'forced'* to satisfied a version of the univalence axiom (for any two object of the category the type of isomorphism between them is equal to the type of equality between two such object). For such categories one has the functoriality along isomorphism of any possible construction (because it boils down to functoriality along equality which is tautological...). So in some sense one can say that any construction that is natural enough to be performed within a version of type theory consistant with univalence should satisfies this Eilenberg-MacLane thesis with respect to isomorphism.
Of course it is not currently well understood to what extent one can develop ordinary mathematics in a way compatible to univalence but it is hoped that this is possible, and that would be in my opinion one of the best possible answer to this question.
*: by 'forced' I mean that they only call categories those which satisfies this univalence axiom. Classical construction on categories generally preserve this property and most usual categories satisfies it. Moreover any category can be replaced by an equivalent categories satisfying this axioms by looking at the essential image of its Yoneda embeddings.