  We present Keck-I MOSFIRE near-infrared spectroscopy for a sample of 13
compact star-forming galaxies (SFGs) at redshift $2\leq z \leq2.5$ 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 $z\sim2$, as measured from
stellar absorption lines. Since log(M$_{\star}$/M$_{\rm{dyn}}$)$=-0.06\pm0.2$
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
$z\gtrsim2$ are already losing gas to become the immediate progenitors of
compact quiescent galaxies by $z\sim2$.
