Gluing pseudo functors via $n$-fold categories

Gluing of two pseudo functors has been studied by Deligne, Ayoub, and others in the construction of extraordinary direct image functors in 'etale cohomology, stable homotopy, and mixed motives of schemes. In this article, we study more generally the gluing of finitely many pseudo functors. Given pseudo functors $F_i\text{\hspace{0.17em}⁣}:{\mathcal{A}}_i\to \mathcal{D}$ defined on sub-$2$-categories ${\mathcal{A}}_i$ of a $2$-category $\mathcal{C}$, we are concerned with the problem of finding pseudo functors $\mathcal{C}\to \mathcal{D}$ extending $F_i$ up to pseudo natural equivalences. With the help of $n$-fold categories, we organize gluing data for $n$ pseudo functors into $2$-categories. We establish general criteria for equivalence between such $2$-categories for $n$ pseudo functors and for $n-1$ pseudo functors, which can be applied inductively to the gluing problem. Results of this article are used in arXiv:1006.3810 to construct extraordinary direct image functors in 'etale cohomology of Deligne-Mumford stacks.