"He will end up dead".
Does the word "dead" denote an adverb here?
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"Dead" in that sentence is an adjective.
Compare to: "He will end up rich" or "He will end up fat".
These are all descriptions of the person in the future.
"will end up" is the verb phrase.
"will" means that you are talking about the future "end" is the verb itself "up" is a preposition which modifies the meaning of "end", and cannot be separated from it. "end up" is actually a phrasal verb where the two parts join together to make a new meaning.
Compare to: Breakout, Break-in, Breakup, Breakdown, Breakthrough, Breakaway, Break-off...
(Note that Breakup and Breakdown are not opposites.)
The adjective dead does not describe the manner of ending. It describes the state that will be reached at the end. You might say "He will end up as a dead man/in a dead state." That's why you here have an adjective in adverb position.
I don't think that complicated linguistic terms help understand such adjective constructions. They are more frequent than one normally thinks. They always occur when they don't describe the manner in which the verb action is performed, but when the describe anything else in reference to the subject as in