Final answer:
The Hudson's Bay Company was the British company that competed directly with France in the fur trade.
Step-by-step explanation:
The British company that competed directly with France in the fur trade was the Hudson's Bay Company. Among the options given, the Hudson's Bay Company was established in 1670 and was granted a monopoly over trade in the lucrative fur-rich regions of what is now Canada. This put the company in direct competition with the French fur trade, which at that time was controlled by the Company of New France. While the British East India Company and the Royal African Company were also significant British trading entities during the same era, they were more involved with trade in Asia and Africa respectively and did not compete with France in the North American fur trade.