Here is a paper I found by Adamek that generalizes Domain theory into categories of categories called Scott Complete Categories. The category of Scott Complete categories is denoted SCC. For years, I had a theory that the category describing quantum theory, Hilb, was a compact element in some domain of categories because Hilb has finitely many axioms. Actually, I am not sure if we even have a categorical model of Hilb that has exactly all the axioms we expect from quantum theory. If you have worked with Hilb, you know that you can see Hilbert spaces as the familiar topological spaces, metric spaces, modules, sets, Banach spaces etc. Right now, I want to say the following: Hilb is a colimit in SCC, and the diagram consists of categories {$A$, $B$...etc}. Is it the case that Hilb can be seen as a colimit in SCC where the diagram of the cone is all the familiar categories I have listed?

Alternatively, Heunen has done work on "compactly accessible categories". In order to get Hilb to be a colimit, do we need to modify Adamek's paper to use the category of compactly accessible categories? Are familiar categories, like those I listed, compactly accessible?