  In the present paper we use the theory of exact completions to study
categorical properties of small setoids in Martin-L\"of type theory and, more
generally, of models of the Constructive Elementary Theory of the Category of
Sets, in terms of properties of their subcategories of choice objects (i.e.
objects satisfying the axiom of choice). Because of these intended
applications, we deal with categories that lack equalisers and just have weak
ones, but whose objects can be regarded as collections of global elements. In
this context, we study the internal logic of the categories involved, and
employ this analysis to give a sufficient condition for the local cartesian
closure of an exact completion. Finally, we apply this result to show when an
exact completion produces a model of CETCS.
