ABSTRACT
This chapter studies the Grothendieck construction on a small tight bipermutative category D with the projection down to D taken into account. This chapter first proves the existence of the non-symmetric Cat-multicategory PFibD with small permutative opfibrations over D as objects. Its multimorphism categories are more subtle and involve the bipermutative structure on D and Laplaza's Coherence Theorem. The non-symmetry of PFibD in general is once again related to the multiplicative braiding in D. The second main result of this chapter lifts the pseudo symmetric Cat-multifunctorial Grothendieck construction to a non-symmetric Cat-multifunctor with codomain PFibD.
