2.3]. Generalizations (under certain assumptions on the C\*-correspondence) were given by Doplicher-Pinzari-Zuccante [@DPZ98 Theorem 3.3], Fowler-Muhly-Raeburn [@FMR03 Theorem 4.1], and Fowler-Raeburn [@FowRae99 Theorem 2.1]. All assumptions were removed by Katsura [@Kat04 Theorem 6.2, Theorem 6.4] by using a sharp analysis of cores and a conceptual argument involving short exact sequences. Another proof for ${{\mathcal{O}}}_X$ was provided by Muhly-Tomforde [@MuhTom04] by using a tail adding technique. A remarkable extension to the much broader class of pre-C\*-correspondences is given by Kwaśniewski [@Kwa11]. The second author [@Kak14] gave an alternative proof of the GIUT that treats all $J$-relative Cuntz-Pimsner algebras with $J \subseteq J_X$: *a representation $(\pi, t)$ of ${}_A X_A$ lifts to a faithful representation of ${{\mathcal{O}}}(J, X)$ if and only if $(\pi,t)$ admits a gauge action, $\pi$ is injective and $J = \{a \in A \mid \pi(a) \in \psi_t({{\mathcal{K}}}(X))\}$*. Consequently if $(\pi,t)$ admits a gauge action and $\pi$ is injective then ${\mathrm{C}^*}(\pi,t) \simeq {{\mathcal{O}}}(J,X)$ for $$J = \{a \in A \mid \pi(a) \in \psi_t({{\mathcal{K}}}(X))\}.$$

We remark that when ${}_A X_A$ is non-degenerate then it suffices to consider just the representations $(\pi, t)$ with $\pi$ non-degenerate. Indeed it is easy to check that $\pi(A)$ carries a c.a.i. for ${\mathrm{C}^*}(\pi,t)$ and we can 