Our goals in this book are twofold. The first is to achieve conceptual simplicity by a judicious choice of axioms in the setting of topos theory. The second is to make sure that our results apply to classical mathematics. To this end we revisit the notion of a well adapted model of SDG, extend it to SDT, and then assume the existence of one such. An application of the existence of such a model to the theory of unfoldings is then given.