What is it called when someone is done with the task that he/she enlisted for?
- They enlist.
- They do what they signed up for.
- They __.
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Completed or finished the task/job?
In the military, you enlist, you do your tour, and then you're discharged.
Alternatively, in non-military terms, something which has served its purpose after enlistment, might be delisted (a word the spell-checker here doesn't seem to appreciate...:-)
I would say they accomplished the task.
to bring to its goal or conclusion; carry out; perform; finish: to accomplish one's mission
The etymology of the word enlist has to do with the making of a compact between one singular party and one collective party composed of members who contract en masse. Any word having to do with divesture, separation, or even nullification will suffice. The word 'fulfill' and its derivatives would be accurate in the context you describe.
The best word to use generally of oneself is resign, but there are a number of alternatives depending on context.
One antonym of enlist is delist, but that would normally have an object which might be reflexive ("I'd like to delist myself"), and as the ODO link shows, isn't normally used of people.
If you have enlisted in the armed forces, then you are demobilised (or "demobbed") at the end of your service; commissioned officers can resign their commission; members can be cashiered if they are discharged dishonourably.
Thesaurus.com has a number of additional possibilities based on resign, not all of which will be entirely suitable all the time:
give up responsibility abandon, abdicate, bail out, bow out, capitulate, cease work, cede, demit, divorce oneself from, drop, drop out, end service, fold, forgo, forsake, give notice, give up the ship, hand in resignation, hand over, hang it up, leave, quit, relinquish, renounce, retire, secede, separate oneself from, sign off, stand aside, stand down, step down, surrender, terminate, throw in the towel, turn over, vacate, waive, walk out, wash hands of, yield