  In this paper we consider the problem of building rich categories of setoids,
in standard intensional Martin-L\"of type theory (MLTT), and in particular how
to handle the problem of equality on objects in this context. Any
(proof-irrelevant) family F of setoids over a setoid A gives rise to a category
C(A, F) of setoids with objects A. We may regard the family F as a setoid of
setoids, and a crucial issue in this article is to construct rich or large
enough such families. Depending on closure conditions of F, the category C(A,
F) has corresponding categorical constructions. We exemplify this with finite
limits. A very large family F may be obtained from Aczel's model construction
of CZF in type theory. It is proved that the category so obtained is isomorphic
to the internal category of sets in this model. Set theory can thus establish
(categorical) properties of C(A, F) which may be used in type theory. We also
show that Aczel's model construction may be extended to include the elements of
any setoid as atoms or urelements. As a byproduct we obtain a natural extension
of CZF, adding atoms. This extension, CZFU, is validated by the extended model.
The main theorems of the paper have been checked in the proof assistant Coq
which is based on MLTT. A possible application of this development is to
integrate set-theoretic and type-theoretic reasoning in proof assistants.
