  Semi-simplicial and semi-cubical sets are commonly defined as presheaves over
respectively, the semi-simplex or semi-cube category. Homotopy Type Theory then
popularized an alternative definition, where the set of n-simplices or n-cubes
are instead regrouped into the families of the fibers over their faces, leading
to a characterization we call indexed. Moreover, it is known that
semi-simplicial and semi-cubical sets are related to iterated Reynolds
parametricity, respectively in its unary and binary variants. We exploit this
correspondence to develop an original uniform indexed definition of both
augmented semi-simplicial and semi-cubical sets, and fully formalize it in Coq.
