Q-system Completion of 2-functors

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.