ABSTRACT
This chapter introduces pseudo symmetric Cat-multifunctors. They preserve the symmetric group action up to coherent natural isomorphisms that satisfy four coherence axioms. They are required to strictly preserve the colored units and the composition. In later chapters it is proved that the Grothendieck construction for each small tight bipermutative category and inverse K-theory are pseudo symmetric Cat-multifunctors. This chapter also proves the existence of a 2-category with small Cat-multicategories as objects, pseudo symmetric Cat-multifunctors as 1-cells, and pseudo symmetric Cat-multinatural transformations as 2-cells. Another theorem proves a local characterization of adjoint equivalences in this 2-category. This concludes Part I.
