crusade vs lost cause

crusade and lost cause both are nouns.

Word	Noun	Adjective	Verb	Adverb
crusade	Yes	No	Yes	No
lost cause	Yes	No	No	No

As nouns, lost cause is a hyponym of crusade; that is, lost cause is a word with a more specific, narrower meaning than crusade:

crusade: a series of actions advancing a principle or tending toward a particular end
lost cause: a defeated cause or a cause for which defeat is inevitable

crusade (noun)	lost cause (noun)
a series of actions advancing a principle or tending toward a particular end	a defeated cause or a cause for which defeat is inevitable

crusade (verb)	lost cause (verb)
go on a crusade; fight a holy war
exert oneself continuously, vigorously, or obtrusively to gain an end or engage in a crusade for a certain cause or person; be an advocate for