Tannaka-Krein duality shows how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to \mathbf{Vect}_{\mathbb{C}}$.

On the other hand, there is Takesaki's theorem which shows how to recover a (separable) $C^*$-algebra $A$ from its representation theory. We have just started looking into the details of this and trying to reformulate this in categorical terms. It seems tantalizingly similar to the Tannaka-Krein reconstruction theorem. In particular, it seems that Takesaki secretly also considers natural transformations from the forgetful functor $F:\mathbf{Rep}(A)\to\mathbf{Hilb}$ to itself.

So, the question is:

Can Takesaki's duality theorem indeed be formulated in categorical terms similar to Tannaka-Krein duality? Where can we read about it?

We're mostly interested in the unital case. Follow-up question:

Consider the category of unital $C^*$-algebras and unital completely positive maps. Is there a "nice" description of its opposite as a concrete category?