  We classify which dual functors on a unitary multitensor category are
compatible with the dagger structure in terms of groupoid homomorphisms from
the universal grading groupoid to $\mathbb{R}_{>0}$ where the latter is
considered as a groupoid with one object. We then prove that all unitary dual
functors induce unitarily equivalent bi-involutive structures. As an
application, we provide the unitary version of the folklore correspondence
between shaded planar ${\rm C^*}$ algebras with finite dimensional box spaces
and unitary multitensor categories with a chosen unitary dual functor and
chosen generator. We make connection with the recent work of Giorgetti-Longo to
determine when the loop parameters in these planar algebras are scalars.
Finally, we show that we can correct for many non-spherical choices of dual
functor by adding the data of a spherical state on
$\operatorname{End}_{\mathcal{C}}(1_{\mathcal{C}})$, similar to the spherical
state for a graph planar algebra.
