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

This requires a sufficiently low pressure. Because the closer to the equator, the higher the temperature (regardless of seasonal changes), and the cyclone near the ground in the northern hemisphere is counterclockwise (geostrophic force), so the north wind blows to the left of the low pressure, blowing the air with high latitude and low temperature to the equator (the equator is in the south of the northern hemisphere), and the cold air collides with the warm air in the south, forming a cold front. On the right, it's a warm front. But for the southern hemisphere, the equator. Because cyclone air is warm in the north and cold in the south, southerly winds blow on the left, blowing high-latitude air to the equator. It's a cold front, on the contrary, it's a warm front on the right. As long as you know the direction of cyclone rotation, imagine which direction the equator is, and compare it with drawing.