For predual categories C and D we establish isomorphisms between opfibrations
representing local varieties of languages in C, local pseudovarieties of
D-monoids, and finitely generated profinite D-monoids. The global sections of
these opfibrations are shown to correspond to varieties of languages in C,
pseudovarieties of D-monoids, and profinite equational theories of D-monoids,
respectively. As an application, we obtain a new proof of Eilenberg's variety
theorem along with several related results, covering varieties of languages and
their coalgebraic modifications, Straubing's C-varieties, fully invariant local
varieties, etc., within a single framework
