Q-system Completion of 2-functors

Abstract

A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category or a C*-2-category. We prove that, for C*-2-categories $\mcal C$ and $\mcal D$, the C*-2-category $\textbf{Fun}(\mcal C, \mcal D)$ of $ * $-$ 2 $-functors, $ * $-$ 2 $-transformations and $ * $-$ 2 $-modifications is Q-system complete, whenever $\mcal D$ is Q-system complete. We use this result to provide a characterisation of Q-system complete categories in terms of $ * $-$ 2 $-functors and to prove that the $ 2 $-category of actions of a unitary fusion category $\mcal C$ on C*-algebras is Q-system complete.