We present a study of the gas kinematics of star-forming galaxies associated with protocluster 4C 23.56 at $z=2.49$ using $0.4$ resolution CO (4-3) data taken with ALMA. Eleven H$alpha$ emitters (HAEs) are detected in CO (4-3), including six HAEs that were previously detected in CO (3-2) at a coarser angular resolution. The detections in both CO lines are broadly consistent in the line widths and the redshifts, confirming both detections. With an increase in the number of spectroscopic redshifts, we confirm that the protocluster is composed of two merging groups with a total halo mass of $log{(M_{rm cl}/M_{odot})} =13.4-13.6$, suggesting that the protocluster would evolve into a Virgo-like cluster ($>10^{14} M_{odot}$). We compare the CO line widths and the CO luminosities with galaxies in other (proto)clusters ($n_{rm gal}=91$) and general fields ($n_{rm gal}=80$) from other studies. The 4C23.56 protocluster galaxies have CO line widths and luminosities comparable to other protocluster galaxies on average. On the other hand, the CO line widths are on average broader by $approx50%$ compared to field galaxies, while the median CO luminosities are similar. The broader line widths can be attributed to both effects of unresolved gas-rich mergers and/or compact gas distribution, which is supported by our limited but decent angular resolution observations and the size estimate of three galaxies. Based on these results, we argue that gas-rich mergers may play a role in the retention of the specific angular momentum to a value similar to that of field populations during cluster assembly, though we need to verify this with a larger number of samples.
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