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.
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