The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two non-trivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subcategories arises from the fact that they provide a very natural notion of model for many of the strongly polymorphic type theories. The nature of these models is that types are interpreted as objects of the topos, and hence as 'sets' in an intuitionistic sense. They in fact provide 'set-theoretic' models in the sense of Reynolds [16], except that the set-theory involved is non-classical. At the end of the paper we indicate briefly how one (only) of the two full subcategories gives rise to a model of the theory of constructions. This sheds some light on how far we can travel with a simple set-theoretic picture of data types. However, our main concern is to present a clean mathematical characterization of the complete subcategories. The basic idea, due to Peter Freyd, remains very simple—it is that the collection of objects orthogonal to any given object is automatically closed under all existing limits, and to look at the objects orthogonal to the subobject classifier in the effective topos. However, one of our original motivations for writing this paper was that we did not quite believe Freyd's identification of this category. During the course of the paper we shall show that the category of (families of) objects in ^orthogonal to Q is the category of (families of) subquotients of N. The chain of reasoning now proceeds by saying that this second category is equivalent to the externalisation of an internal category, and that the internal category is therefore small complete. Somewhat to our surprise, it was this second stage, trivial for an internal category in Sfe&, that we found difficulty in making precise, and it is in the discussion of notions of equivalence and completeness for internal categories and fibrations that much of the content of the paper now lies. We should warn the reader that even so our treatment is not exhaustive. We discuss only two notions (weak and strong) of equivalence for internal or fibred categories, and only two notions of completeness. Our definitions are welladapted for our present purposes, corresponding more or less to the distinction between being told for any instance of a given set of parameters that there is an object with certain properties, and being given a function that produces one. Let C be a small full subcategory of a topos (see § 3 for details) which is closed under arbitrary products. Then C cannot contain Q, the object of truth values (in fact C cannot contain any object into which Q embeds)—for if A is the disjoint union of the sets in C, then Q^eC, and hence Q^y^A, which contradicts one of the constructive versions of Cantor's theorem. This argument runs parallel