  It is well-known in universal algebra that adding structure and equational
axioms generates forgetful functors between varieties, and such functors all
have left adjoints. The category of elementary doctrines provides a natural
framework for studying algebraic theories, since each algebraic theory can be
described by some syntactic doctrine and its models are morphism from the
syntactic doctrine into the doctrine of subsets. In this context, adding
structure and axioms to a theory can be described by a morphism between the two
corresponding syntactic doctrines, and the forgetful functor arises as
precomposition with this last morphism. In this work, given any morphism of
elementary doctrines, we prove the existence of a left adjoint of the functor
induced by precomposition in the doctrine of subobjects of a Grothendieck
topos.
