  Tambara functors are an equivariant generalization of rings that appear as
the homotopy groups of genuine equivariant commutative ring spectra. In recent
work, Blumberg and Hill have studied the corresponding algebraic structures,
called bi-incomplete Tambara functors, that arise from ring spectra indexed on
incomplete $G$-universes. In this paper, we answer a conjecture of Blumberg and
Hill by proving a generalization of the Hoyer--Mazur theorem in the
bi-incomplete setting. Bi-incomplete Tambara functors are characterized by
indexing categories which parameterize incomplete systems of norms and
transfers. In the course of our work, we develop several new tools for studying
these indexing categories. In particular, we provide an easily checked,
combinatorial characterization of when two indexing categories are compatible
in the sense of Blumberg and Hill.
