  Q-systems are unitary versions of Frobenius algebra objects which appeared in
the theory of subfactors. In recent joint work with R. Hern\'andez Palomares
and C. Jones, the authors defined a notion of Q-system completion for C*/W*
2-categories, which is a unitary version of a higher idempotent completion in
the spirit of Douglas--Reutter and Gaiotto--Johnson-Freyd. In this article, we
prove that Q-system completion is a dagger 3-functor on the dagger 3-category
of C*/W* 2-categories. We also prove that Q-system completion satisfies a
universal property analogous to the universal property satisfied by idempotent
completion for 1-categories.
