The Stacks project

When applied to the property P = 'surjective', this creates a new definition of surjective for non-representable morphisms, which a priori is different than the definition 'surjective on topological spaces' (albeit 03MF shows that the definitions are the same afterall). Is it possible that similar ambiguities arise for other types of morphisms? A similar remark applies to the corresponding place discussing morphisms of algebraic stacks.

I don't have a good suggestion though how to modify the statement of the preceding lemma to ensure such compatibilities for non-representable morphisms (without being super-formal about it, e.g., listing all such properties one-by-one and pointing to relevant lemmas that show the equivalence of the new definition; albeit even with this approach there is no guarantee that the reader won't come up with some new crazy property P, check that it's etale-local on the source and base, and then apply this definition to create a duplicate a priori differing version of what P is). If one were formal about this, one solution would be to restrict the scope of this definition to some explicit list of P's.

So, I think your comment is incorrect, just because "surjective" is not local on the source-and-target. See my comment #481. When writing these definitions I tried to be very careful that this kind of issue would never come up.