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We conduct a 12C16O(J = 1−0) (hereafter CO) mapping survey of 64 galaxies in the Fornax cluster using the Atacama Large Millimeter/submillimeter Array Morita array in cycle 5. CO emission is detected from 23 out of the 64 galaxies. Our sample includes dwarf, spiral, and elliptical galaxies with stellar masses of M star ∼ 106.3−11.6 M ⊙. The achieved beam size and sensitivity are 15″ × 8″ and ∼12 mJy beam−1 at the velocity resolution of ∼10 km s−1, respectively. We study the cold gas (molecular and atomic gas) properties of 38 subsamples with M star > 109 M ⊙ combined with literature H i data. We find that (1) the low star formation (SF) activity in the Fornax galaxies is caused by the decrease in the cold gas mass fraction with respect to stellar mass (hereafter, gas fraction) rather than the decrease of the SF efficiency from the cold gas; (2) the atomic gas fraction is more heavily reduced than the molecular gas fraction of such galaxies with low SF activity. A comparison between the cold gas properties of the Fornax galaxies and their environmental properties suggests that the atomic gas is stripped tidally and by the ram pressure, which leads to the molecular gas depletion with an aid of the strangulation and consequently SF quenching. Preprocesses in the group environment would also play a role in reducing cold gas reservoirs in some Fornax galaxies.