I guess Walter is right. At least in the case of an initial object, it is the case that having a multi-initial object is having an initial object in each slice. What I hadn't thought through was the fact that if a category has a multi-initial object and has a terminal object, then it has an initial object. So having an initial object in each slice does not imply having an initial object (whereas for most properties P, locally P implies P). There is a minor problem with duality. While it is true that a category with a multi-terminal object has a terminal object in each slice, that is also true of a category without a multi-terminal object. So you want that for every co-slice. Michael =====================================================================