Lallement functor is a weak right multiadjoint

For a plural signature $\mathrm{\Sigma }$ and with regard to the category $\mathsf{N}\mathsf{P}\mathsf{I}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }{)}_{\mathsf{s}}$, of naturally preordered idempotent $\mathrm{\Sigma }$-algebras and surjective homomorphisms, we define a contravariant functor ${\mathrm{L}\mathrm{s}\mathrm{y}\mathrm{s}}_{\mathrm{\Sigma }}$ from $\mathsf{N}\mathsf{P}\mathsf{I}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }{)}_{\mathsf{s}}$ to $\mathsf{C}\mathsf{a}\mathsf{t}$, the category of categories, that assigns to $\mathbf{I}$ in $\mathsf{N}\mathsf{P}\mathsf{I}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }{)}_{\mathsf{s}}$ the category $\mathbf{I}$-$\mathsf{L}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma })$, of $\mathbf{I}$-semi-inductive Lallement systems of $\mathrm{\Sigma }$-algebras, and a covariant functor $(\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }){\downarrow }_{\mathsf{s}}\cdot )$ from $\mathsf{N}\mathsf{P}\mathsf{I}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }{)}_{\mathsf{s}}$ to $\mathsf{C}\mathsf{a}\mathsf{t}$, that assigns to $\mathbf{I}$ in $\mathsf{N}\mathsf{P}\mathsf{I}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }{)}_{\mathsf{s}}$ the category $(\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }){\downarrow }_{\mathsf{s}}\mathbf{I})$, of the coverings of $\mathbf{I}$, i.e., the ordered pairs $(\mathbf{A},f)$ in which $\mathbf{A}$ is a $\mathrm{\Sigma }$-algebra and $f\text{\hspace{0.17em}⁣}:\mathbf{A}⟶\mathbf{I}$ a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories ${\int }^{\mathsf{N}\mathsf{P}\mathsf{I}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }{)}_{\mathsf{s}}}{\mathrm{L}\mathrm{s}\mathrm{y}\mathrm{s}}_{\mathrm{\Sigma }}$ and ${\int }_{\mathsf{N}\mathsf{P}\mathsf{I}\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }{)}_{\mathsf{s}}}(\mathsf{A}\mathsf{l}\mathsf{g}(\mathrm{\Sigma }){\downarrow }_{\mathsf{s}}\cdot )$; define a functor ${\mathfrak{L}}_{\mathrm{\Sigma }}$ 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.