 In any model of NFU, a structure which looks just like one of these models can be constructed in the isomorphism classes of well-founded extensional relations. The theory of isomorphism classes of well-founded extensional relations with a top element looks like the theory of (an initial segment of) the usual cumulative hierarchy, because every set in Zermelo-style set theory is uniquely determined by the isomorphism type of the restriction of the membership relation to its transitive closure. The surprise is that we not only see a structure which looks like an initial segment of the cumulative hierarchy: we also see an external endomorphism of this structure which moves a rank (and therefore cannot be a set), in terms of which we can replicate the model construction above and get an interpretation of NFU of this kind inside NFU! The endomorphism is induced by the map \(T\) which sends the isomorphism type of a relation \(R\) to the isomorphism type of \(R^{\iota} = \{ \langle \{x\}, \{y\}\rangle \mid xRy\}\). There is no reason to believe that \(T\) is a function: it sends any relation \(R\) to a relation \(R^{\iota}\) which is one type higher in terms of TST. It is demonstrable that \(T\) on the isomorphism types of well-founded extensional relations is not a set function (we will not show this here, but our discussion of the Burali-Forti paradox below should give a good idea of the reasons for this). See Holmes (1998) for the full discussion.