  For single-sorted algebras, Fujiwara defined, through the concept of family
of basic mapping-formulas, a notion of morphism which generalizes the ordinary
notion of homomorphism between algebras and an equivalence relation, the
conjugation, on the families of basic mapping-formulas, which corresponds to
the relation of inner isomorphism for algebras. In this paper we extend the
theory of Fujiwara about morphisms to the many-sorted algebras, by defining the
concept of polyderivor between many-sorted signatures, which assigns to basic
sorts, words and to formal operations, families of derived terms, and under
which the standard signature morphisms, the basic mapping-formulas of Fujiwara,
and the derivors of Goguen-Thatcher-Wagner are subsumed. Then, by means of the
homomorphisms between B\'enabou algebras, which are the algebraic counterpart
of the finitary many-sorted algebraic theories of B\'enabou, we define the
composition of polyderivors from which we get a corresponding category,
isomorphic to the category of Kleisli for a monad on the standard category of
many-sorted signatures. Next, by defining the notion of transformation between
polyderivors, we endow the category of many-sorted signatures and polyderivors
with a structure of 2-category. From this we get a derived 2-category of
many-sorted specifications in which we prove the equivalence of the many-sorted
specifications of Hall and B\'enabou, and deduce the equivalence of the
categories of Hall and B\'enabou algebras. Besides, by defining corresponding
categories of generalized many-sorted terms, we prove that the realization of
these terms in the many-sorted algebras is invariant under polyderivors and
compatible with the transformations between polyderivors, and from this we get
an example, among others, of the new concept of 2-institution, itself an strict
generalization of that of institution by Goguen and Burstall.
