  We contribute to the knowledge of the quantifier completions and their
applications by using the language of doctrines. This algebraic presentation
allows us to properly analyse the behaviour of the existential and universal
quantifiers. We wish to convey the following points: the first is that these
completions preserve the lattice structure and the distributive lattice
structure of the fibres under opportune hypotheses which turn out to be
preserved as well; the second regards the applications, in particular to the
dialectica construction; the third is that these free constructions carry on
some relevant choice principles.
