  We report on the production of a novel cold mixture of fermionic $^{53}$Cr
and $^{6}$Li atoms delivered by two Zeeman-slowed atomic beams and collected
within a magneto-optical trap (MOT). For lithium, we obtain clouds of up to $4
\,10^8$ atoms at temperatures of about $500\,\mu$K. A gray optical molasses
stage allows us to decrease the gas temperature down to $45(5)\,\mu$K. For
chromium, we obtain MOTs comprising up to $1.5\, 10^6$ atoms. The availability
of magnetically trappable metastable $D$-states, from which $P$-state atoms can
radiatively decay onto, enables to accumulate into the MOT quadrupole samples
of up to $10^7$ $^{53}$Cr atoms. After repumping $D$-state atoms back into the
cooling cycle, a final cooling stage decreases the chromium temperature down to
$145(5)\,\mu$K. While the presence of a lithium MOT decreases the lifetime of
magnetically trapped $^{53}$Cr atoms, we obtain, within a 5 seconds duty cycle,
samples of about $4\, 10^6$ chromium and $1.5\,10^8$ lithium atoms. Our work
provides a crucial step towards the production of degenerate Cr-Li Fermi
mixtures.
