fox hunter vs equestrian

fox hunter and equestrian both are nouns.

Word	Noun	Adjective	Verb	Adverb
fox hunter	Yes	No	No	No
equestrian	Yes	Yes	No	No

As nouns, equestrian is a hypernym of fox hunter; that is, equestrian is a word with a broader meaning than fox hunter:

Other hypernyms of fox hunter include horseback rider.

fox hunter (noun)	equestrian (noun)
a mounted hunter who follows the hounds in pursuit of a fox	a person skilled in equitation

fox hunter (adjective)	equestrian (adjective)
	of or relating to or featuring horseback riding
	of or relating to or composed of knights