The category of points of has objects . In other words, the objects are the split epimorphisms of with a choice of a splitting, or equivalently the retracts in . At the level of morphisms it is just a bit more complex than the morphisms in each , namely the slice and coslice triangles become squares. More precisely, a morphism is a pair of morphisms , in such that and .
The fibration of points is the codomain-assigning functor , , . It is a fibered category in the sense of Grothendieck. Its fibers are which are (as mentioned above) pointed and finitely complete. A morphism (in notation as above) is cartesian] iff are the sides of a pullback square in (i.e. is a pullback of along and a pullback of along ). The inverse image functor for this fibration is exactly described by the rule above.