Let S be the set of scalings {n −1:n=1,2,3,...} and let L z =z Z 2, z∈S, be the corresponding set of scaled lattices in R 2. In this paper averaging operators are defined for plaquette functions on L z to plaquette functions on L z′ for all z′, z∈S, z′=dz, d∈{2,3,4,...}, and their coherence is proved. This generalizes the averaging operators introduced by Balaban and Federbush. There are such coherent families of averaging operators for any dimension D=1,2,3,... and not only for D=2. Finally there are uniqueness theorems saying that in a sense, besides a form of straightforward averaging, the weights used are the only ones that give coherent families of averaging operators.