Canonical pre-3-Hilbert structure on functor 2-categories

Construct a canonical pre-3-Hilbert space structure on the dagger 2-category Fun^\dagger(\fX → \fY) of UAF-preserving dagger 2-functors from a pre-3-Hilbert space \fX to a 3-Hilbert space \fY such that the following are equivalent: (1) a 2-functor F is an isometric equivalence; (2) there exist isometric equivalences F^*F ⇒ id_\fX and id_\fY ⇒ FF^*, where F^* is the unitary adjoint of F.

Background

The authors introduce isometric equivalence between pre-3-Hilbert spaces and note the existence of unitary adjoints for functors between 2-Hilbert spaces. They conjecture an enrichment of the 2-categorical functor space with a pre-3-Hilbert structure that aligns the abstract notion of isometric equivalence with the existence of isometric unit–counit data for unitary adjunctions.

Realizing this conjecture would endow Fun^\dagger(\fX → \fY) with a UAF and a compatible spherical weight, providing a robust functional-analytic framework for higher-categorical mapping objects in the 3-Hilbert setting and enabling Yoneda-style results at the 3-Hilbert level.