  The modulation of the valley structure in two-dimensional valley materials is
vital in the field of valleytronics. The multiferroicity provides possibility
for multiple modulations of the valley, including the magnetic and electric
means. Based on the first-principle calculations, we study the valley
properties and associated manipulations of multiferroic Co$_2$CF$_2$ monolayers
with different stacking patterns. Our calculations show that the Co$_2$CF$_2$
monolayer in the H$^{\prime}$ phase is a ferrovalley material, with sizable
valley splittings. By rotating the magnetization direction, the valley
splittings can be tuned for both the magnitude and sign. The electric field,
driving the reversal of the electric polarization, can also change the
magnitude of the valley splittings. Besides, a metastable T$^{\prime}$ phase
exhibits valley splittings as well, of which the magnitude and sign can be
simultaneously controlled by applied magnetic and electric fields. These
findings offer a practical way for realizing highly tunable valleys by
multiferroic couplings.
