Abstract We use the complete Sydney-AAO Multi-object Integral field spectrograph (SAMI) Galaxy Survey to determine the contribution of slow rotators, as well as different types of fast rotators, to the stellar mass function of galaxies in the local Universe. We use stellar kinematics not only to discriminate between fast and slow rotators, but also to distinguish between dynamically cold systems (i.e., consistent with intrinsic axis ratios<0.3) and systems including a prominent dispersion-supported bulge. We show that fast rotators account for more than $80\%$ of the stellar mass budget of nearby galaxies, confirming that their number density overwhelms that of slow rotators at almost all masses from 109 to 1011.5M⊙. Most importantly, dynamically cold disks contribute to at least $25\%$ of the stellar mass budget of the local Universe, significantly higher than what is estimated from visual morphology alone. For stellar masses up to 1010.5M⊙, this class makes up $>=30\%$ of the galaxy population in each stellar mass bin. The fact that many galaxies that are visually classified as having two-components have stellar spin consistent with dynamically cold disks suggests that the inner component is either rotationally-dominated (e.g., bar, pseudo-bulge) or has little effect on the global stellar kinematics of galaxies.