We present Keck-I MOSFIRE near-infrared spectroscopy for a sample of 13 compact star-forming galaxies (SFGs) at redshift $2leq z leq2.5$ with star formation rates of SFR$sim$100M$_{odot}$ y$^{-1}$ and masses of log(M/M$_{odot}$)$sim10.8$. Their high integrated gas velocity dispersions of $sigma_{rm{int}}$=230$^{+40}_{-30}$ km s$^{-1}$, as measured from emission lines of H$_{alpha}$ and [OIII], and the resultant M$_{star}-sigma_{rm{int}}$ relation and M$_{star}$$-$M$_{rm{dyn}}$ all match well to those of compact quiescent galaxies at $zsim2$, as measured from stellar absorption lines. Since log(M$_{star}$/M$_{rm{dyn}}$)$=-0.06pm0.2$ dex, these compact SFGs appear to be dynamically relaxed and more evolved, i.e., more depleted in gas and dark matter ($<$13$^{+17}_{-13}$%) than their non-compact SFG counterparts at the same epoch. Without infusion of external gas, depletion timescales are short, less than $sim$300 Myr. This discovery adds another link to our new dynamical chain of evidence that compact SFGs at $zgtrsim2$ are already losing gas to become the immediate progenitors of compact quiescent galaxies by $zsim2$.
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