  The Schur orthogonality relations are a cornerstone in the representation
theory of groups. We utilize a generalization to weak Hopf algebras to provide
a new, readily verifiable condition on the skeletal data for deciding whether a
given bimodule category is invertible and therefore defines a Morita
equivalence. As a first application, we provide an algorithm for the
construction of the full skeletal data of the invertible bimodule category
associated to a given module category, which is obtained in a unitary gauge
when the underlying categories are unitary. As a second application, we show
that our condition for invertibility is equivalent to the notion of
MPO-injectivity, thereby closing an open question concerning tensor network
representations of string-net models exhibiting topological order. We discuss
applications to generalized symmetries, including a generalized Wigner-Eckart
theorem.
