We measure the spectral reflectivity of a Cat (so named because its first conjugated image was that of a cat) self-pumped conjugator. We find that the transmission and the phase-conjugate reflectivity of the conjugator always vary periodically with wavelength and that the periodicity is caused by a resonator loop that springs up and self-adjusts inside the crystal. Depending on the spectral width of the incident beam, the conjugator will have either a minimum or a maximum reflectivity at the center of the writing spectrum. We show that the sign of the coherence function after one round trip of the internal resonator determines the spectral behavior of the conjugator, and we experimentally map out the dependence of the conjugator’s reflectivity and transmission on the spectral content of the incident beam.