  We report on simultaneous sub-Doppler laser cooling of fermionic $^6$Li and
$^{40}$K using the D$_1$ optical transitions. We compare experimental results
to a numerical simulation of the cooling process applying a semi-classical
Monte Carlo wavefunction method. The simulation takes into account the three
dimensional optical molasses setup and the dipole interaction between atoms and
the bichromatic light field driving the D$_1$ transitions. We discuss the
physical mechanisms at play, we identify the important role of coherences
between the ground state hyperfine levels and compare D$_1$ and D$_2$
sub-Doppler cooling. In 5 ms, the D$_1$ molasses phase largely reduces the
temperature for both $^6$Li and $^{40}$K at the same time, with a final
temperature of 44 $\mu$K and 11 $\mu$K, respectively. For both species this
leads to a phase-space density close to $10^{-4}$. These conditions are well
suited to directly load an optical or magnetic trap for efficient evaporative
cooling to quantum degeneracy.
