You have collapsed two very different ideas into one.
When I say that concepts are formed by minds in response to reality, I do not mean that concepts are arbitrary or private. Rather, it means they are
formed, as opposed to being
invented without constraint (made the fuck up).
But now you've lost the ability to claim that my five and your five have anything to do with each other. I agree with all your other examples of fiveness, days in a week, fingers on a hand. But a 3 fingered man may simply come along, and confidently hold up his 3-fingered hand, and say "Here is my five". Where is your basis of reference to tell him otherwise?
I dont see how you're going to tell the 3-fingered man that your five is better than his five without taking for granted the existence of something whose essence exists nowhere in an empiricist's reality.
Your three-fingered man does not show what you think it shows. A strange commonality in this thread so far!
If that man holds up three fingers and calls it "five", he is not presenting an alternative but equally valid "five", he is making a mistake. The fact that we can identify it as a mistake in the first place already answers your question about a basis of reference.
The basis is not "my five vs your five", but the
actual quantity.
Three is not five. That is not a matter of perspective, brain state, or preference. It is a matter of how many units there are. The concept "five" refers to a specific quantitative relationship. Misapplying the word does not create a new instance of that relationship.
That is, a disagreement, which your example shows, does not undermine objectivity. It does the opposite, it
presupposes objectivity. You cannot call something wrong unless there is something it is wrong
about.
The same point applies to your earlier concern about "whose brain". Different people can form the same concept because they are responding to the same features of reality. The concept is in the mind, what the concept refers to is not. That is why communication, correction, and agreement are possible in the first place.
Now on the morality point...
justification for concepts like morality. There are scant few true atheists in the west, most of them go about their days appealing to these things without a second thought to justification, but if you put their feet to the fire their justification will ultimately contain something like Platonism.
"Either arbitrary mental constructions or transcendent abstract entities" is a false dichotomy.
Appealing to a separate realm of "transcendentals" does not justify anything, it just relocates the problem. Much like OP, you still have to explain how you know these entities exist, how you identify their content, and why they should guide action. Simply asserting that there is a realm where "fiveness" or "goodness" or "faggotness" exists does not supply those answers. If anything, it just makes it harder to explain, because now you have detached the standard from the reality it's supposed to apply to.
The cleaner approach is the same as before.
Concepts, including moral ones, are formed in response to real features and relations in the world. Their validity depends on whether they correctly identify and integrate those features, not on whether they correspond to a separate metaphysical layer. So you don't get justification by multiplying entities. Just the opposite, you end up losing it, because you replace a checkable relation to reality with an uncheckable assertion about something beyond it.
In addition, you get minus points for sidestepping the question I asked you. In what you said, you did not identify any missing explanatory power. The three-fingered man is just an error case, not a gap in abstraction. The assertion that disagreement undermines objectivity actually presupposes a standard. And you jumped to morality without showing why abstraction can't ground it. None of that has answered my question.
To reiterate: What, specifically, does abstractionism fail to account for that requires transcendentals?