Answer:
For global supremacy. They fought over colonial territories because they believed they're the rightful leaders over these territories.
Step-by-step explanation:
The map mentioned Seven Years war and the United States Independence.
Seven Years war was a war between France and Britain over ownerships of the lands in Caribbean region. These lands contain a lot of sugar cane that was regarded as a hot commodity at that time.
During United States, the France formed an alliance with the colonists against British empire. The French believed that united states were a major income for the British government. Helping them to be independent from Britain will weaken the British' economy. This will be beneficial for France in their competition against British empire.