Let κ be a regular cardinal. We study Gabriel-Ulmer duality when one
restricts the 2-category of locally κ-presentable categories with
κ-accessible right adjoints to its locally full sub-2-category of
κ-presentable Grothendieck topoi with geometric κ-accessible
morphisms. In particular, we provide a full understanding of the locally full
sub-2-category of the 2-category of κ-small cocomplete categories with
κ-colimit preserving functors arising as the corresponding 2-category of
presentations via the restriction. We analyse the relation of these
presentations of Grothendieck topoi with site presentations and we show that
the 2-category of locally κ-presentable Grothendieck topoi with
geometric κ-accessible morphisms is a reflective sub-bicategory of the
full sub-2-category of the 2-category of sites with morphisms of sites
genearated by the weakly κ-ary sites in the sense of Shulman [37].Comment: 25 page