Dagger compact closed categories are commonly said to be an abstraction of Hilbert spaces and is suppose to capture concepts such as unitary maps, scalars, basis, inner products. See for example the below Wikipedia article and the introduction in the paper by Peter Selinger.
https://en.wikipedia.org/wiki/Dagger_compact_category
https://ncatlab.org/nlab/files/SelingerPositiveMaps.pdf
In particular, let $\mathcal{C}$ be a dagger compact closed category with tensor unit $I$. Let $A$ be another object in $\mathcal{C}$. Then we can consider morphisms in $\textrm{Hom}(I, A)$ to be the vectors in the "Hilbert space". If $v,w \in \textrm{Hom}(I,A)$ then we can define the "inner-product" as: $$\langle v, w \rangle_{\mathcal{C}} = v^{\dagger}w.$$ In the case of $\mathcal{C} = \textrm{Hilb}$ this construction returns back the original Hilbert space and inner-product. However, of course, in general this does not give a Hilbert space as a priori $\textrm{Hom}(I,A)$ is not even a vector space and $\langle \cdot, \cdot \rangle_{\mathcal{C}}$ is not a true inner-product.
Now this is my question.
For a dagger compact closed category $\mathcal{C}$ (possibly requiring further restrictions on $\mathcal{C}$), does there exist a general procedure to build a Hilbert space using the data of $(\textrm{Hom}(I,A), \langle \cdot, \cdot \rangle_{\mathcal{C}})$ ? Or more generally, is there a way to obtain a collection of Hilbert spaces from $\mathcal{C}$?