Left adjoint to precomposition in elementary doctrines

Abstract: 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.