No problem with Mike's remark that "locally P" should mean "P holds in every slice". This is for a property of a category. When using "locally right adjoint" almost two decades ago I asked myself whether the func- tor A/a --> B/Fa induced by F:A-->B (and an A-object a) is an accep- table notion of "slice of the functor F", decided that would be okay, and therefore used the name. Diers' notion of multi-adjointness gives more than just right adjointness of the slices A/a --> B/Fa : there is a smallness condition on the number of non-isomorphic "local units", and these are F-epic. Hence it would be confusing to use the name in a more general sense now. I may be wrong in assuming that the notion of "slice of a functor" as above is acceptable. Comments welcome. Walter Tholen. =====================================================================