In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between Gal(L/K) and C_m/H where H is generated by the split primes in L). Does this mean:

1)in category theory terms: there is a natural transformation between the functors from abelian extensions over K to abelian groups given by Gal(/K) and C_m/H (where H_ is generated by the primes split over _/K).

2)or some kind of vaguer statement about whether we need to make choices along the definition of the map.

or maybe 2) is precisely encoded in the definition of 1).