In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.
The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.
Wiki's definitions seem to be inconsistent. What's a structure which does not have an underlying set?
Does that mean a structure can arise on its own, or have a underlying class instead, like a category in which both objects and morphisms are actually proper classes and not sets?