A factorization system (E, M) on a category A gives rise to the covariant category-valued pseudofunctor P of A sending each object to its slice category over M. This article characterizes the P so obtained as follows: their object images have terminal objects, and they admit bicategorically cartesian liftings, up to equivalence, of slice-category projections. It is clear that, and how, (E, M) can be recovered from such a P. The correspondence thus described is actually the second of three similar ones between certain (E, M) and certain P that the article presents. In the first one the characterization of the P has all ultimately bicategorical ingredients replaced with their categorical analogues. A category A with such a P is precisely what the author has called a `slicing site'. In the article's terms the associated (E, M) are again factorization systems - but the concept conveyed extends the standard one by not obliging isomorphisms to belong to either factor class -, namely those that are `right semireplete' (isomorphisms do belong to M and `left semistrict' (morphisms in M are monic relative to E). The third correspondence subsumes the other two; here the (E, M) are all right-semireplete factorization systems.
Keywords: factorization system, slice categories, cartesian morphisms, slicing site
2010 MSC: 18D99, 18A32
Theory and Applications of Categories, Vol. 30, 2015, No. 14, pp 489-526.