  Over the last few years, $ab~initio$ methods have become an increasingly
popular tool to evaluate intrinsic carrier transport properties in 2D
materials. The lack of experimental information, and the progress made in the
development of DFT tools to evaluate electronic band structures, phonon
dispersions, and electron-phonon scattering matrix-elements, have made them a
favored choice. However, a large discrepancy is observed in the literature
among the $ab~initio$ calculated carrier mobility in 2D materials. Some of the
discrepancies are a result of the physical approximations made in calculating
the electron-phonon coupling constants and the carrier mobility. These
approximations can be avoided by using a sophisticated transport model.
However, despite using appropriate transport models, the uncertainty in the
reported carrier mobility is still quite large in some materials. The major
differences observed between these refined model calculations are the `flavors'
of DFT (exchange-correlation functional, pseudopotential, and the effect of
spin-orbit coupling) used. Here, considering several monolayer 2H-TMDs as
examples, we calculate the low- and high-field transport properties using
different `flavors' of DFT, and calculate a range for the electron mobility
values. We observe that in some materials the values differ by orders of
magnitude (For example, in monolayer WS$_{2}$ the electron low-field mobility
varies between 37 cm$^{2}$/(V$\cdot$s) and 767 cm$^{2}$/(V$\cdot$s)). We
analyze critically these discrepancies, and try to understand the limitations
of the current $ab~initio$ methods in calculating carrier transport properties.
