In set theory with the axiom of foundation, the \(\in\)-relations on transitive classes are up to isomorphism just the extensional well-founded relations. In the absence of the axiom of foundation one may require various axioms of universality (e.g. that every binary relation (which is extensional) has an homomorphism (an isomorphism, resp.) onto the \(\in\)-relation on a transitive set or class). The authors discuss several such axioms as realizations of a ''free construction principle'' and establish their mutual relationship within the framework of Gödel-Bernays set theory.