  In this paper we take up again the deformation theory for $K$-linear
pseudofunctors initiated in a previous work (Adv. Math. 182 (2004) 204-277). We
start by introducing a notion of a 2-cosemisimplicial object in an arbitrary
2-category and analyzing the corresponding coherence question, where the
permutohedra make their appearence. We then describe a general method to obtain
cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in
the 2-category ${\bf Cat}_K$ of small $K$-linear categories and prove that the
deformation complex introduced in the above mentioned work can be obtained by
this method from a 2-cosemisimplicial object that can be associated to the
pseudofunctor. Finally, using a generalization to the context of $K$-linear
categories of the deviation calculus introduced by Markl and Stasheff for
$K$-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to
the integrability of an $n^{th}$-order deformation of a pseudofunctor indeed
correspond to cocycles in the third cohomology group, a question which remained
open in our previous work.
