Lallement functor is a weak right multiadjoint ...

For a plural signature $Σ$ and with regard to the category $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$, of naturally preordered idempotent $Σ$-algebras and surjective homomorphisms, we define a contravariant functor $\mathrm{Lsys}_Σ$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, the category of categories, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $\mathbf{I}$-$\mathsf{LAlg}(Σ)$, of $\mathbf{I}$-semi-inductive Lallement systems of $Σ$-algebras, and a covariant functor $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \cdot)$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \mathbf{I})$, of the coverings of $\mathbf{I}$, i.e., the ordered pairs $(\mathbf{A},f)$ in which $\mathbf{A}$ is a $Σ$-algebra and $f\colon \mathbf{A}\longrightarrow \mathbf{I}$ a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the ... : 31 pages. arXiv admin note: text overlap with arXiv:2305.03581 ...