MM Chalmers: Schaffer on Furnishing Functions

In a part of ‘Ontological Anti-Realism’ which I didn’t comment on (§§8-11), David Chalmers considers an objection against anti-realism based on the idea that the absolute unrestricted quantifier has an objective, determinate semantic value. I don’t want to assess his response to the objection here (see related discussion here, and references there).

In order to analyse existence assertions, however, he tentatively introduces the notion of a furnished world—an ordered pair of a world and a domain—and a furnishing function—a mapping from worlds to domains—(see the end of §8).

In his comments to the paper, Jonathan Schaffer objects:

The argument for heavyweight realism about fundamental structure: Furnishing functions are maps from a world to a domain. But a function is a map from one structure (‘the input’) to another (‘the output’). One cannot have a well-defined function without there being some articulated structure to the input. In particular we must be able to specify the arguments of the function. Any function is either complete or partial. It is either injective or not. It is either surjective or not. None of these classifications would make sense unless the input (‘the world’) already comes with some fundamental articulated structure inbuilt, to feed into the function. … I conclude that the framework that Chalmers actually supplies is at least half-realist, in the sense that it presupposes heavyweight realism about fundamental structure. (pp. 2-3)

I am probably missing something here. For I understood that a furnishing function was a map from the class of worlds to the class of domains, whose arguments were precisely just worlds. Thus I don’t see why there being such mappings requires in any sense any “articulated structure” in the items to which the function is applied. Can anyone help?