hair dye vs dye

hair dye and dye both are nouns.

Word	Noun	Adjective	Verb	Adverb
hair dye	Yes	No	No	No
dye	Yes	No	Yes	No

As nouns, dye is a hypernym of hair dye; that is, dye is a word with a broader meaning than hair dye:

hair dye: a dye for coloring the hair
dye: a usually soluble substance for staining or coloring e.g. fabrics or hair

Other hypernyms of hair dye include dyestuff.

hair dye (noun)	dye (noun)
a dye for coloring the hair	a usually soluble substance for staining or coloring e.g. fabrics or hair

hair dye (verb)	dye (verb)
	color with dye
	color for microscopic study