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The Sagittarius dwarf spheroidal galaxy (Sgr dSph) is in an advanced stage ofdisruption but still hosts its nuclear star cluster (NSC), M54, at its center.In this paper, we present a detailed kinematic characterization of the threestellar populations present in M54: young metal-rich (YMR); intermediate-agemetal-rich (IMR); and old metal-poor (OMP), based on the spectra of $\sim6500$individual M54 member stars extracted from a large MUSE/VLT dataset. We findthat the OMP population is slightly flattened with a low amount of rotation($\sim0.8$ km s$^{-1}$) and with a velocity dispersion that follows a Plummerprofile. The YMR population displays a high amount of rotation ($\sim5$ kms$^{-1}$) and a high degree of flattening, with a lower and flat velocitydispersion profile. The IMR population shows a high but flat velocitydispersion profile, with some degree of rotation ($\sim2$ km s$^{-1}$). Wecomplement our MUSE data with information from \textit{Gaia DR2} and confirmthat the stars from the OMP and YMR populations are comoving in 3D space,suggesting that they are dynamically bound. While dynamical evolutionaryeffects (e.g. energy equipartition) are able to explain the differences invelocity dispersion between the stellar populations, the strong differences inrotation indicate different formation paths for the populations, as supportedby an $N$-body simulation tailored to emulate the YMR-OMP system. This studyprovides additional evidence for the M54 formation scenario proposed in ourprevious work, where this NSC formed via GC accretion (OMP) and in situformation from gas accretion in a rotationally supported disc (YMR).

A deep view into the nucleus of the Sagittarius Dwarf Spheroidal Galaxy with MUSE. II. Kinematic characterization of the stellar populations