Here's my take on the objectivity of morality.
If morality is objective , ie, built into the fabric of the world around us (rather than just in our minds), then it is necessary that the world around us is the product of a greater mind.
What makes you think objectivity requires that a thing be built into the fabric of the world around us rather than just in our minds? Schizophrenia is not built into the fabric of the world around us.
As far as I can tell, my best empirical models include the diagnosis of schizophrenia as highly predictive of certain future observations, given past observations, so it is as built in to the fabric of the world as any other bit of ontology at the level of macroscopic description.
1) It is an objective fact
about Smith's mind that he prefers wine to beer.
2) It is an objective fact
about Smith's mind that he values having $3 less than he values having a gallon of gasoline in his car.
3) It is an objective fact
about Smith's mind that he has symptoms of schizophrenia.
1') It is an objective fact that Smith says, "wine
is better than beer."
2') It is an objective fact that Smith says, "gas
is worth at least $3."
3') It is an objective fact that Smith says, "the CIA
is sending me messages through my dental work."
1'') It is not an objective fact whether or not wine
is better than beer.
2'') It is not an objective fact that gasoline has more than or less than $3 worth of labor-value "stored up" in it.
3'') But there is, in fact, an objective fact of the matter about whether or not there is information transfer from a certain government agency to a certain person's mouth.
One of these things is not like the other. When a realist disagrees with an antirealist over the mind-dependence of, say, monetary value, they are disagreeing over the truth of sentences like 2'', not 2 or 2'.
There is the (philosophically) trivial or banal claim that cars are dependent on minds, since cars are causally consequent on beings with minds. But it would be highly non-trivial -- indeed, it would be earth-shattering -- to learn that cars depend on minds in the sense that whether or not the sentence "cars exist" was true depended on whether the speaker believed in cars, or had a negative attitude towards cars. Just as it would be highly non-trivial to learn (as a current of 20th century thought around such figures as Foucault and Szasz actually once asserted!) that schizophrenia qua medical diagnosis is culturally relative, and simply a way of society's elites to delegitimize alternate ways of thinking by calling it a "sickness".