  We determine the existential completion of a primary doctrine, and we prove
that the 2-monad obtained from it is lax-idempotent, and that the 2-category of
existential doctrines is isomorphic to the 2-category of algebras for this
2-monad. We also show that the existential completion of an elementary doctrine
is again elementary. Finally we extend the notion of exact completion of an
elementary existential doctrine to an arbitrary elementary doctrine.
