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 $\text{mcal}C$ and $\text{mcal}D$, the C*-2-category $\mathbf{\text{Fun}}(\text{mcal}C,\text{mcal}D)$ of $\ast$-$2$-functors, $\ast$-$2$-transformations and $\ast$-$2$-modifications is Q-system complete, whenever $\text{mcal}D$ is Q-system complete. We use this result to provide a characterisation of Q-system complete categories in terms of $\ast$-$2$-functors and to prove that the $2$-category of actions of a unitary fusion category $\text{mcal}C$ on C*-algebras is Q-system complete.