The hyperconnected maps that are local

A level j:Ej→E of a topos E is said to have monic skeleta if, for every X in E, the counit j!(j⁎X)→X is monic. For instance, the centre of a hyperconnected geometric morphism is such a level. We establish two related sufficient conditions for an adjunction to extend to a level with monic skeleta. As an application, we characterize the hyperconnected geometric morphisms that are local providing an interesting expression for the associated centres that suggests a generalization of open subtoposes. As a corollary, we obtain that a hyperconnected p:E→S is pre-cohesive if and only if p⁎:E→S preserves coequalizers and p⁎:S→E is cartesian closed.