  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 $F_i\colon \mathcal{A}_i\to \mathcal{D}$ 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.
