The relativist plausibly says, “contradictory statements can be true.” The objectivist says, “contradictory statements cannot both be true.” Here, we have mutually incompatible statements, if the objectivist is correct. The relativist says, “no, you are wrong, they can.” The objectivist says, “you have just proven my point!” And he has.
How? Well, their beliefs imply the negation of the other and the relativist cannot, on pains of being extremely incoherent, say that the objectivist is correct and that he himself is correct. As a matter of fact, the relativist does disagree with the objectivist, thereby cutting his only limb upon which to precariously perch. Perhaps this reduces, then, to subjectivism: “I’m right for me and you’re right for you, let’s just all get along.” The trouble with this is, at least, practical. I don’t care what you think is right or wrong, true or false, except in extremely simple and uninteresting cases. Normal persons, in normal situations, care about what is true simpliciter. Saying what you think is not saying what is true (unless one is God).
If the relativist holds to their guns after such a conversation and insists that, no, the objectivist is still wrong and they are correct, the only solution is to say, “have a nice lonely life.”