I can prove to you that all of chemistry is subjective.
Assume that a chemist were to make the statement, "Carbon atoms have six protons."
I have already proved that chemistry is subjective. This is because no scientist can give any type of objective argument showing that six proton atoms have to be called 'carbon' and cannot have any other name. The decision to call six proton atoms 'carbon' is arbitrary and subjective. So, chemistry itself is subjective.
There is a second argument, along these same lines, yielding the same general conclusion.
Not only is it the case that the scientist cannot objectively prove that six proton atoms must be called 'carbon' - that this choice is not arbitrary - but can give no objective answer to the question of why he has decided to write about 6-proton atoms.
There are over a hundred different atoms that the chemist could be talking about - hydrogen, oxygen, iron, uranium. The decision to talk about carbon atoms, as opposed to one of these other types of atoms - is entirely arbitrary. It is up to the whim of the chemist what atoms he is going to write about. That is to say, the choice is totally subjective. Therefore, chemistry is totally subjective.
In fact, both of these accusations about chemistry are true. Yet, neither of them prove that chemistry itself is subjective. The objectivity of chemistry is not the least bit threatened by the arbitrariness of language or the arbitrariness of what any individual chooses to write about.
In the realm of morality, one of the things I choose to write about are the relationships between malleable desires and other desires. I claim that the degree to which a malleable desire fulfills other desires relates to the degree that others have for using social tools such as praise and condemnation to mold those desires. People generally (not individually, but generally) have many and strong reasons to use these social tools to promote malleable desires that tend to fulfill other desires, and to inhibit malleable desires that tend to thwart other desires.
The question then comes up, "Why are you talking about these relationships between malleable desires and all other desires? You can give me no objective answer to the question of why you choose to talk about these and not some other desires and relationships. Because you cannot give an answer to this question, this proves that morality is subjective rather than objective."
Yet, this is no more of a threat to objective morality than the claim that the chemist chooses to talk about carbon rather than oxygen threatens objective chemistry. What makes a field of study objective or subjective is not the author's preferences with respect to what part of reality he wishes to study, but whether the claims he makes about that object of study are objectively true or false.
This is the challenge that somebody who wants to claim that desire utilitarianism (desirism) is not compatible with the claim that morality is objective. It's not a valid objection to say that the agent has a choice as to what aspect of reality he is going to study. One must demonstrate that he is saying things about that object of study that are not objectively true.
In my case, the person raising the objection must be able to point to something that I have said about the relationship between malleable desires and other desires and say, "Well, that's just your assertion. It is not a fact about those desires. We can make just as much sense of the real world if we deny that claims that you have made."
This is quite different from the objection that says, "You have selected to write about relationships between malleable desires and other desires. You could have selected to write about the history of Great Britain or the chemical composition of stars. Because you made an arbitrary choice as to what to write about, your claims are not objectively true."
If a person wants to make a meaningful objection they have to talk about the claims that I am making about things, not my choices as to what things to talk about or what to call them. That is the type of objection that will challenge any theory.
So, what have I said about the relationships between malleable desires and other desires that is not objectively true?