In this chapter, the tricategory of bicategories is presented in full detail. After a preliminary discussion of the whiskerings of a lax transformation with a lax functor, the chapter goes on to define a tricategory. The rest of the chapter proves in detail the existence of a tricategory with small bicategories as objects (i.e. a tricategory of bicategories), pseudofunctors as 1-cells, strong transformations as 2-cells, and modifications as 3-cells.
This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.