Is this Frobenius Monad an internal category in [Hilb, Hilb]? In this paper  we see a Frobenius Monad in example 5.2.  Suppose we take Hilb as the underlying category.  Is this Frobenius Monad an internal category in [Hilb, Hilb]?  If you can show that it is an internal category, please give some data about that category.
Heunen and Tull have a vaguely similar result in theorem 4.7.
 A: Let me just expand on one aspect of my comment and recall the old chestnut that "most" bicategories whose objects are some kind of "structure" tend to fall into one of two classes:


*

*Bicategories whose 1-morpisms are "maps", e.g: 


*

*sets, functions, and identity 2-morphisms

*rings, ring homomorphisms, and identity 2-morphisms

*categories, functors, and natural transformations

*...



versus


*Bicategories whose 1-morphisms are "bimodules", e.g:


*

*sets, spans, and maps of spans

*sets, relations, and inclusion of relations

*rings, bimodules, and bimodule homomorphisms

*categories, profunctors, and natural transformations

*...



Very often, a bicategory of type 1 is included into a bicategory of type 2 to form a proarrow equipment, e.g. 


*

*(sets and functions) $\subset$ (sets and spans)

*(sets and functions) $\subset$ (sets and relations)

*(rings and homomorphisms) $\subset$ (rings and bimodules)

*(categories and functors) $\subset$ (categories and profunctors)

*...


A monad in a type 1 bicategory often looks like, well, a monad. A monad in a type 2 bicategory often looks like some kind of category. Exercise: work out what a monad is in each of the above bicategories. In particular, a monad in $Span(C)$ is the same as an internal category in $C$.
The question is asking for a comparison between monads in a type 1 bicategory ($Cat$) and monads in a type 2 bicategory ($Span([Hilb,Hilb])$), which doesn't look very promising.
The linked comparison by Heunen and Tull compares monads in two type-2 bicategories (a category of relations vs. a category of spans), so passes the smell test.
