  In this follow-up to [16], we continue developing the notion of a lego
category and its many applications to stratifiable spaces and the computation
of their Grothendieck classes. We illustrate the effectiveness of this
construction by giving very short derivations of the class of a quotient by the
"stratified action" of a discrete group [1], the class of a crystallographic
quotient, the class of both a polyhedral product and a polyhedral (or
simplicial) configuration space [8], the class of a permutation product [19]
and, foremost, the class of spaces of $0$-cycles [11].
