The Topos theory example is completely wrong though? > For example, if E is the topos of sets, then the slice category is the category of pointed sets: the obje

The Topos theory example is completely wrong though? > For example, if E is the topos of sets, then the slice category is the category of pointed sets: the objects are then sets together with a function selecting a point as a “base point”. There's no object X where E/X is pointed sets. If you take the co-slice of a one point set 1\E that would be the category of pointed sets. The right way to think about E/X is that its objects are "families of sets indexed by X", because a function f : Y -> X defines for each x in X a set f^{-1}(x). If you know about dependent types, this is basically a type dependent on X.