Why is the trough in the low-pressure weather system cold on the left and warm on the right in both hemispheres?

The cold air from the northern part of the northern hemisphere is deflected to the right by the geostrophic bias, so the cold air in the west of the cyclone is stronger, resulting in a cold front near the warm air. The warm air from the south is turned to the right, so there is a strong heating force in the east of the cyclone, which leads to a warm front near the cold air.

The cold air from the southern part of the southern hemisphere is biased to the left by the geostrophic bias, so the cold air in the western part of the cyclone is stronger, resulting in a cold front near the warm air. The warm air from the north is turned to the left, so there is a strong heating force in the east of the cyclone, which leads to a warm front near the cold air.