  In this paper, we discuss the generalization of finitary $2$-representation
theory of finitary $2$-categories to finitary birepresentation theory of
finitary bicategories. In previous papers on the subject, the classification of
simple transitive $2$-representations of a given $2$-category was reduced to
that for certain subquotients. These reduction results were all formulated as
bijections between equivalence classes of $2$-representations. In this paper,
we generalize them to biequivalences between certain $2$-categories of
birepresentations. Furthermore, we prove an analog of the double centralizer
theorem in finitary birepresentation theory.
