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This second part comes to the construction of the spectrum associated to asituation of multi-adjunction. Exploiting a geometric understanding of itsmulti-versal property, the spectrum of an object is obtained as the spaces oflocal units equipped with a topology provided by orthogonality aspects. Afterrecalling Diers original construction, this paper introduces new material.First we explain how the situation of multi-adjunction can be corrected in asituation of adjunction between categories of modeled spaces as in thetopos-theoretic approach. Then we come to the 2-functorial aspects of theprocess relatively to a 2-category of Diers contexts. We propose anaxiomatization of the notion of spectral duality through morphisms betweenfibrations over a category of spatial objects, and show how such situations getback right multi-adjoint functors.

On Diers theory of Spectrum II: Geometries and dualities