Change of base, Cauchy-completeness and reversibility

We investigate the effect on Cauchy complete objects of the change of base 2-functor V-Cat ! W-Cat induced by a two-sided enrichment V ! W. We restrict our study to the case of locally partially ordered bases. The reversibility notion introduced in [Wal82] is extended to two-sided enrichments and Cauchy completion. We show that a reversible left adjoint two-sided enrichment F : V ! W between locally partially ordered reversible bicategories induces an adjunction F a F : V-SkCRcCat * W-SkCRcCat between sub-categories of skeletal and Cauchy-reversible complete enrichments. We give two applications: sheaves over locales and group actions.