In this final chapter we go back to basics and treat several different categorical aspects of the theory of approach spaces. First, we establish the existence of an infinite collection of stable subcategories of the category approach spaces. Second, we construct a quasi-topos supercategory of the category of approach spaces which will serve as a starting-point for the construction of all the hulls. We then construct the extensional topological hull, the quasi-topos hull and the cartesian closed topological hull. Finally we give a new proof of a lax-algebraic description of the category of approach spaces.