  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 $10^{9}$ to $10^{11.5}{\rm M_\odot}$. 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 $10^{10.5}{\rm M_\odot}$, 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.
