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Diers developed a general theory of right multi-adjoint functors leading to apurely categorical, point-set construction of spectra. Situations ofmultiversal properties return sets of canonical solutions rather than a uniqueone. In the case of a right multi-adjoint, each object deploys a canonical coneof local units jointly assuming the role of the unit of an adjunction. Thisfirst part revolves around the theory of multi-adjoint and recalls or precisesresults that will be used later on for geometric purpose. We also study theweaker notion of local adjoint, proving Beck-Chevalley conditions relatinglocal adjunctions and the equivalence with the notion of stable functor. Wealso recall the link with the free-product completion, and describefactorization aspects involved in a situation of multi-adjunction. The relationbetween accessible right multi-adjoints and locally finitely multipresentablecategories is also revisited.

On Diers theory of Spectrum I : Stable functors and right multi-adjoints