  We provide a new description of Joyal's arithmetic universes through a
characterization of the exact and regular completions of pure existential
completions. We show that the regular and exact completions of the pure
existential completion of an elementary doctrine $P$ are equivalent to the
$\mathsf{reg}/\mathsf{lex}$ and $\mathsf{ex}/\mathsf{lex}$-completions,
respectively, of the category of predicates of $P$. This result generalizes a
previous one by the first author with F. Pasquali and G. Rosolini about
doctrines equipped with Hilbert's $\epsilon$-operators. Thanks to this
characterization, each arithmetic universe in the sense of Joyal can be seen as
the exact completion of the pure existential completion of the doctrine of
predicates of its Skolem theory. In particular, the initial arithmetic universe
in the standard category of ZFC-sets turns out to be the completion with exact
quotients of the doctrine of recursively enumerable predicates.
