Lallement Functor is a Weak Right Multiadjoint

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For a plural signature Σ and with regard to the category NPIAlg(Σ)s, of naturally preordered idempotent Σ-algebras and surjective homomorphisms, we define a contravariant functor LsysΣ from NPIAlg(Σ)s to Cat, the category of categories, that assigns to I in NPIAlg(Σ)s the category I-LAlg(Σ), of I-semi-inductive Lallement systems of Σ-algebras, and a covariant functor (Alg(Σ)↓s·) from NPIAlg(Σ)s to Cat, that assigns to I in NPIAlg(Σ)s the category (Alg(Σ)↓sI), of the coverings of I, i.e., the ordered pairs (A,f) in which A is a Σ-algebra and a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories ∫NPIAlg(Σ)sLsysΣ and ∫NPIAlg(Σ)s(Alg(Σ)↓s·); define a functor LΣ from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.