After defining, for each many-sorted signature
Σ = (S, Σ), the category
Ter(
Σ), of generalized terms for
Σ (which is the dual of the Kleisli category for
\mathbb
TS{\mathbb {T}_{\bf \Sigma}}, the monad in
Set
S
determined by the adjunction
TS \dashv G
S{{\bf T}_{\bf \Sigma} \dashv {\rm G}_{\bf \Sigma}} from
Set
S
to
Alg(
Σ), the category of
Σ-algebras), we assign, to a signature morphism
d from
Σ to
Λ, the functor
dà{{\bf d}_\diamond} from
Ter(
Σ) to
Ter(
Λ). Once defined the mappings that assign, respectively, to a many-sorted signature the corresponding category of generalized
terms and to a signature morphism the functor between the associated categories of generalized terms, we state that both mappings
are actually the components of a pseudo-functor Ter from
Sig to the 2-category
Cat. Next we prove that there is a functor Tr
Σ, of realization of generalized terms as term operations, from
Alg(
Σ) ×
Ter(
Σ) to
Set, that simultaneously formalizes the procedure of realization of generalized terms and its naturalness (by taking into account
the variation of the algebras through the homomorphisms between them). We remark that from this fact we will get the invariance
of the relation of satisfaction under signature change. Moreover, we prove that, for each signature morphism
d from
Σ to
Λ, there exists a natural isomorphism
θ
d from the functor Tr
L °(Id ×
dà){{{\rm Tr}^{\bf {\bf \Lambda}} \circ ({\rm Id} \times {\bf d}_\diamond)}} to the functor Tr
S °(
d* ×Id){{\rm Tr}^{\bf \Sigma} \circ ({\bf d}^* \times {\rm Id})}, both from the category
Alg(
Λ) ×
Ter(
Σ) to the category
Set, where
d* is the value at
d of the arrow mapping of a contravariant functor Alg from
Sig to
Cat, that shows the invariant character of the procedure of realization of generalized terms under signature change. Finally,
we construct the many-sorted term institution by combining adequately the above components (and, in a derived way, the many-sorted
specification institution), but for a strict generalization of the standard notion of institution.
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