We prove, by using the concept of schematic interpretation, that the natural embedding from
the category ISL, of intuitionistic sentential pretheories and i-congruence classes of
morphisms, to the category CSL, of classical sentential pretheories and c-congruence
classes of morphisms, has a left adjoint, which is related to the double negation
interpretation of Gödel-Gentzen, and a right adjoint, which is related to the Law of
Excluded Middle. Moreover, we prove that from the left to the right adjoint there is a
pointwise epimorphic natural transformation and that since the two endofunctors at CSL,
obtained by adequately composing the aforementioned functors, are naturally isomorphic to the
identity functor for CSL, the string of adjunctions constitutes an adjoint cylinder. On
the other hand, we show that the operators of Lindenbaum-Tarski of formation of algebras from
pretheories can be extended to equivalences of categories from the category CSL,
respectively, ISL, to the category Bool, of Boolean algebras, respectively,
Heyt, of Heyting algebras. Finally, we prove that the functor of regularization from
Heyt to Bool has, in addition to its well-known right adjoint (that is, the
canonical embedding of Bool into Heyt) a left adjoint, that from the left to the
right adjoint there is a pointwise epimorphic natural transformation, and, finally, that such a
string of adjunctions constitutes an adjoint cylinder.