  We give a characterization of a finite-dimensional commuting square of
C*-algebras with a normalized trace that produces a hyperfinite type II_1
subfactor of finite index and finite depth in terms of Morita equivalent
unitary fusion categories. This type of commuting squares were studied by N.
Sato, and we show that a slight generalization of his construction covers the
fully general case of such commuting squares. We also give a characterization
of such a commuting square that produces a given hyperfinite type II_1
subfactor of finite index and finite depth. These results also give a
characterization of certain 4-tensors that appear in recent studies of matrix
product operators in 2-dimensional topological order.
