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\u03a3<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> from \n \n $$\\textsf {NPIAlg}(\\Sigma )_{\\textsf {s}}$$<\/jats:tex-math>\n \n \n NPIAlg<\/mml:mi>\n \n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to \n \n $$\\textsf {Cat}$$<\/jats:tex-math>\n \n Cat<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the category of categories, that assigns to \n \n $${\\textbf {I}}$$<\/jats:tex-math>\n \n I<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in \n \n $$\\textsf {NPIAlg}(\\Sigma )_{\\textsf {s}}$$<\/jats:tex-math>\n \n \n NPIAlg<\/mml:mi>\n \n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> the category \n \n $${\\textbf {I}}$$<\/jats:tex-math>\n \n I<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-\n \n $$\\textsf {LAlg}(\\Sigma )$$<\/jats:tex-math>\n \n \n LAlg<\/mml:mi>\n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, of \n \n $${\\textbf {I}}$$<\/jats:tex-math>\n \n I<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-semi-inductive Lallement systems of \n \n $$\\Sigma $$<\/jats:tex-math>\n \n \u03a3<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-algebras, and a covariant functor \n \n $$(\\textsf {Alg}(\\Sigma )\\,{\\downarrow _{\\textsf {s}}}\\, \\cdot )$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n Alg<\/mml:mi>\n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n \n \n \u2193<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n \n \u00b7<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> from \n \n $$\\textsf {NPIAlg}(\\Sigma )_{\\textsf {s}}$$<\/jats:tex-math>\n \n \n NPIAlg<\/mml:mi>\n \n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to \n \n $$\\textsf {Cat}$$<\/jats:tex-math>\n \n Cat<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, that assigns to \n \n $${\\textbf {I}}$$<\/jats:tex-math>\n \n I<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in \n \n $$\\textsf {NPIAlg}(\\Sigma )_{\\textsf {s}}$$<\/jats:tex-math>\n \n \n NPIAlg<\/mml:mi>\n \n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> the category \n \n $$(\\textsf {Alg}(\\Sigma )\\,{\\downarrow _{\\textsf {s}}}\\, {\\textbf {I}})$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n Alg<\/mml:mi>\n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n \n \n \u2193<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n \n I<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, of the coverings of \n \n $${\\textbf {I}}$$<\/jats:tex-math>\n \n I<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, i.e., the ordered pairs \n \n $$({\\textbf {A}},f)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n A<\/mml:mi>\n ,<\/mml:mo>\n f<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in which \n \n $${\\textbf {A}}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a \n \n $$\\Sigma $$<\/jats:tex-math>\n \n \u03a3<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-algebra and \"Equation missing\" a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories \n \n $$\\int ^{\\textsf {NPIAlg}(\\Sigma )_{\\textsf {s}}}\\textrm{Lsys}_{\\Sigma }$$<\/jats:tex-math>\n \n \n \n \u222b<\/mml:mo>\n \n NPIAlg<\/mml:mi>\n \n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n \n Lsys<\/mml:mtext>\n \u03a3<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n $$\\int _{\\textsf {NPIAlg}(\\Sigma )_{\\textsf {s}}}(\\textsf {Alg}(\\Sigma )\\,{\\downarrow _{\\textsf {s}}}\\, \\cdot )$$<\/jats:tex-math>\n \n \n \n \u222b<\/mml:mo>\n \n NPIAlg<\/mml:mi>\n \n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n Alg<\/mml:mi>\n \n (<\/mml:mo>\n \u03a3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n \n \n \u2193<\/mml:mo>\n <\/mml:mrow>\n s<\/mml:mi>\n <\/mml:msub>\n \n \u00b7<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>; define a functor \n \n $$\\mathfrak {L}_{\\Sigma }$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n \u03a3<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> 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. 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