  For a plural signature $\Sigma$ and with regard to the category
$\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$, of naturally preordered idempotent
$\Sigma$-algebras and surjective homomorphisms, we define a contravariant
functor $\mathrm{Lsys}_{\Sigma}$ from $\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ to
$\mathsf{Cat}$, the category of categories, that assigns to $\mathbf{I}$ in
$\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ the category
$\mathbf{I}$-$\mathsf{LAlg}(\Sigma)$, of $\mathbf{I}$-semi-inductive Lallement
systems of $\Sigma$-algebras, and a covariant functor
$(\mathsf{Alg}(\Sigma)\,{\downarrow_{\mathsf{s}}}\, \cdot)$ from
$\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ to $\mathsf{Cat}$, that assigns to
$\mathbf{I}$ in $\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ the category
$(\mathsf{Alg}(\Sigma)\,{\downarrow_{\mathsf{s}}}\, \mathbf{I})$, of the
coverings of $\mathbf{I}$, i.e., the ordered pairs $(\mathbf{A},f)$ in which
$\mathbf{A}$ is a $\Sigma$-algebra and $f\colon \mathbf{A}\longrightarrow
\mathbf{I}$ a surjective homomorphism. Then, by means of the Grothendieck
construction, we obtain the categories
$\int^{\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}}\mathrm{Lsys}_{\Sigma}$ and
$\int_{\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}}(\mathsf{Alg}(\Sigma)\,{\downarrow_{\mathsf{s}}}\,
\cdot)$; define a functor $\mathfrak{L}_{\Sigma}$ 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
P{\l}onka functor and the Lallement functor.
