Categorical approach to rigidity of Roe-like algebras of coarse spaces

We demonstrate that any full and faithful $\ast$-functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such $\ast$-functors between approximable categories and prove that the functor associating each locally finite coarse space with its approximable category is full and faithful.