$\begingroup$

I believe that there is no definition of canonical which covers all of its uses in mathematics. Heuristic evidence for this is that the definition of a canonical map in Wikipedia is vague.

The word appears in number theory and in some cases it can be made rigorous, and in other cases it is more of a heuristic issue.

Sometimes people say "canonical isomorphism" when they just mean "natural (in the sense of category theory) isomorphism".

In algebraic geometry there is a convention where canonically isomorphic objects are declared equal and the $=$ symbol is used to denote the canonical isomorphism. Strictly speaking this is abuse of the $=$ sign but it has never seemed to cause any problems in algebraic geometry. To give an explicit example: if $R$ is a commutative ring and $M$ is an $R$-module then there is a canonical identification of the abelian groups $M_f:=M\otimes_R R[1/f]$ and $M_g:=M\otimes_R R[1/g]$ when the set of primes containing $f$ equals the set of primes containing $g$ -- both of these are the value on the open subset $D(f)=D(g)\subseteq Spec(R)$ of the quasicoherent sheaf associated to the $R$-module $M$. In EGA1, (1.3.3) Grothendieck says that the map between these modules is "un homomorphisme canonique fonctoriel" and then uses the $=$ symbol and writes $M_f=M_g$. They are not equal in the strict sense, but they both satisfy the same universal property so as long as mathematicians only ever do things to these modules which can be done using the universal property -- and this is exactly what they do -- then this abuse of the $=$ symbol will not cause problems. But one could in theory argue that many many uses of the $=$ symbol in EGA and elsewhere in algebraic geometry should really say "is canonically isomorphic to". In Milne's book on etale cohomology he explicitly admits this, saying at the beginning of his book that canonical isomorphisms will be denoted $=$, without ever explicitly saying what a canonical isomorphism is. Here, what appears to be going on is that the isomorphisms are sufficiently natural that every conceivable sensible diagram that one can come up with will commute. Note that there is an extra subtlety here -- $A = B$ is usually in mathematics a true-false statement. In Milne's book it is sometimes actually implicitly encoding a map from $A$ to $B$, and the usual proof that equality is an equivalence relation is being beefed up to assertions of the form "composing two canonical isomorphisms gives a canonical isomorphism" and so on. This phenomenon occurs all over the place and is very well-hidden. However there is no doubt that this convention "works".

Finally, in the Langlands philosophy one can find assertions saying that the local or global Langlands correspondences are canonical. This is a use of the term which I find more unsettling because the situation seems to be that people believe that there is one "special" correspondence, and they can list a bunch of properties which it should have, but nobody has proved the theorem that there is at most one correspondence with these properties and indeed it may well not be true that the current list of properties that we have defines the correspondence uniquely. One can try and wriggle out of this by asking that the correspondence commutes with functoriality, however functoriality is a big open question and the statement of functoriality is not completely understood in full generality (what happens for non-classical groups at bad primes). My current thinking about this use of the word is that it is more expressing a hope that one day in the future, people will figure out what we were talking about. However if one works with general connected reductive groups, I am pretty sure that this day is not yet here. It is for me a very interesting usage of the word.

Finally it might be worth mentioning that even in the case of GL_1, there are two canonical Langlands correspondences! Class field theory is the statement that two abelian groups are canonically isomorphic, but these groups contain elements of order bigger than 2, and so one can apply inversion on one side but not the other and change an isomorphism into a different one. They are distinguished by having different names -- one is "the canonical isomorphism sending uniformisers to arithmetic Frobenii" and the other is "the canonical isomorphism sending uniformisers to geometric Frobenii".