Hyperrigidity of C*-correspondences

We show that hyperrigidity for a C*-correspondence $(A,X)$ is equivalent to non-degeneracy of the left action of the Katsura ideal ${\mathcal{J}}_X$ on $X$. Due to the work of Katsoulis and Ramsey, our result shows that if $G$ is a locally compact group acting on $(A,X)$ and the Katsura ideal ${\mathcal{J}}_X$ acts on $X$ non-degenerately then the Hao-Ng isomorphism problem for reduced crossed products has a positive solution and the Hao-Ng isomorphism problem for full crossed products has a partial solution.