The present paper studies compatible adjacency relations for digital products such as a C-compatible adjacency (or the L C-property in [21]), an S-compatible adjacency in [27] (or the L S-property in [21]), which are used to study product properties of digital images. Furthermore, to study an automorphism group of a Cartesian product of two digital coverings which do not satisfy a radius 2 local isomorphism, which remains open, the paper uses some properties of an ultra regular covering in [24]. By using this approach, we can substantially classify digital products.