That’s not sufficient—there can be wildly different, incompatible universalizable morality systems based on different premises and axioms; and each could reasonably claim to be that they are a true morality and the other is a tribal shibboleth.
As an example (but there are others), many of the major religious traditions would definitely claim to be universalizable systems of morality; and they are contradicting each other on some points.
OK, for a slightly clearer example, in the USA abortion debate, the pro-life “camp” definitely considers pro-life to be moral and wants to apply to everyone; and pro-choice “camp” definitely considers pro-choice to be moral and to apply to everyone.
This is not a symbolic point, it is a moral question that defines literally life-and-death decisions.