Mathematicians use the term "without loss of generality" in a proof when they pick one case to work with and want you to know the choice doesn’t matter, because the same argument would still work for any other case.
So, if a statement works without loss of generality, it should work for more than one "thing": more than one set, more than one variable, more than one map, or, *more than one type of person*. We want the culture of research mathematics to work the same way.