Relationship of Repc(C) to bicartesian-closed and topos closures

Determine how the representation functor Repc from a concrete category C of structured objects relates to CCartisain(Repc(C)) and CTopos(Repc(C)), defined as the minimal bicartesian closed subcategory and minimal topos subcategory of the ambient Token category PC that contain Repc(C).

Background

In the conclusion, the authors introduce operators CCartisain and CTopos on subcategories of Token categories, intended to produce the minimal bicartesian closed and topos subcategories containing a given subcategory. They then pose a specific problem about the categorical relationship between the representation Repc(C) and these closures.

Resolving this relationship would clarify how structured-object categories embed into Token Space and what categorical properties (closure, limits, exponentials, subobject classification) are preserved or reflected by Repc.