Needs More Examples.
In some works, there's an antagonist who isn't known for being friendly with the protagonists. But sometimes, in situations when one of said protagonists (often the main/lead character, but supporting characters qualify as well) exhibits a heroic or otherwise positive character trait that sufficiently impresses the antagonist, the latter will not be above commending them or otherwise showing his approval of them or their character trait in question. It will particularly stand out if, previously, the antagonist was hostile or (more usually) simply dismissive of the targeted protagonist.
The action may usually be performed by the Noble Demon, but bonus points if it's the Big Bad delivering the praise. Note that the commendation is sincere, and is not an attempt by the villain to screw with the hero's head. The villain may also do it because he and the hero are Not So Different after all. It may also be a Pet the Dog moment to show the villain isn't such a bad guy after all (even if hewon't admit it).
A Lawful Evil character, or a character with Evil Virtues, may have this as part of his/her personality, especially in a situation where Villains Never Lie. This can also be a trait of the Friendly Enemy (though depending on the work, it probably won't reach Go-Karting with Bowser levels). The Antagonist in Mourning may also do this as a tribute to the fallen hero (whether the hero is actually dead or not).
May frequently overlap with Combat Compliment and Worthy Opponent, where the praise is given during a fight. Compare Villain Takes an Interest, where the villain may become interested in being a mentor to the hero, and compare and contrast We Can Rule Together, where the villain tries to convince the hero that they would do better working together instead of against each other.
Contrast Your Approval Fills Me with Shame, where a villain praises the hero for a negative trait or action and the hero feels disgusted as a result. Also different from Baddie Flattery, wherein the villain is still sincere in his praise of a positive trait but the praise comes across as utterly creepy and disturbing (though this might be a sister trope to that one). And, again, note the second paragraph above.
Please note: The character to whom the praise is given is generally a true-blue hero. If the Anti-Hero or the Anti-Villain is the recipient of the praise, it's a bit trickier since the former has villainous characteristics while skirting the line between good and evil, and the latter has heroic characteristics while still not being a clearly good guy.
Anime and Manga
The titular character of Kenichi: The Mightiest Disciple has gotten this from several antagonists, including three members of the Yami group's One Shadow Nine Fists martial artists and at least one member of Yami's disciple group YOMI.
Agaard Jum Sai, the One Shadow Nine Fists' Muay Tai master and the Rival Turned Evil to Ryouzapaku's Apachai Hopachai, is the first noted example among the One Shadow Nine Fists. Shortly after Kenichi defeats Tirawat Koukin, Agaard's student and a member of YOMI, Agaard commends Kenichi on his tenacity and praises his ability as Apachai's disciple, while at the same time telling Koukin there's no shame in losing to such a skilled opponent. Plus he gives Kenichi pointers on defending himself against a Master-class weapons user moments afterward, himself unable to help directly due to being temporarily paralyzed from his fight with Apachai.
The second example is "God Fist" Akira Hongo, the Rival Turned Evil to Ryozanpakou's Sakaki Shio. When he momentarily derides his student Sho Kanou's Redemption Equals Death sacrifice to protect Miu during the D of D Tournament arc, Kenichi angrily declares that Sho's sacrifice was not a waste and he won't allow Hongo to say it was (interestingly, insulting Sho's memory is in fact a Berserk Button for Hongo). The exchange between them goes this way:
Hongo: (Sho) committed a mistake...and it is my responsibility that the softness inside of him was not totally erased.
Kenichi:THAT'S WRONG! He...that time, I thought that he was really incredible! As a person and as a man! That's why I don't want you to say that his last action was some kind of mistake!!!
Kenichi: Ryouzanpaku first disciple...Shirahama Kenichi.
The third example is the One Shadowa.k.a. SaigaFurinji himself, who commends Kenichi for his determination in protecting Miu, and later flashes his gauntlets to Kenichi from a distance as a mark of respect.
One Shadow: A man like you is worth trusting.
As recently as Chapter 500, the newest member of YOMI Berserker expresses acknowledgment of Kenichi's fighting prowess even as they begin their skirmish at the amusement park where Kenichi and his friends had gone to relax (meanwhile, the other YOMI members are fighting Kenichi's friends).
Berserker: Indeed, you're quite good. No-one has been able to dodge my attacks so perfectly after I trained as a YOMI, so far. I was right in choosing you as my target, Kenichi Shirahama.
In One Piece, Luffy gets this from at least four major figures over the course of his journey.
The first is Dracule Mihawk, who as a member of the Shichibukai is technically an enemy of all pirates but who proves himself to be somewhat of a Friendly Enemy to the Straw Hat crew and to Luffy and Zoro in particular (Luffy due to Mihawk's past association with the former's idol Shanks, Zoro due to Mihawk being impressed with his tenacity as a swordsman).
Mihawk: Youngster, what is your goal?
Luffy: Pirate King!
Mihawk:(smirks) That is a path of incomparable danger. Even more difficult than to surpass me.
Luffy: I don't care. I will become one anyway!
The second is Enel, the Big Bad of the Skypiea arc, when Luffy declares that he is going to be Pirate King (this example could also be a Combat Compliment since they're in the middle of a fight at the time).
Enel: Pirate King? What country does he rule?
Luffy: He is the king of the sea!
Enel: That doesn't sound half bad...Let's finish this in the sky.
In the Kaizoku fan-sub airing of the anime's version of that scene:
Luffy: He (Pirate King) is ruler of the entire ocean!
Enel: I see. A laudable title.
The third is Rob Lucci, during the Enies Lobby arc, who commends Luffy on his leadership skills and tells him that he (Luffy) is a better leader than Lucci's boss Spandam (who, to be fair, is an idiot).
The fourth is Blackbeard, who commends and encourages Luffy to pursue his dreams no matter what anyone else says. Notable because this happens both before and after Luffy finds out who the person actually is, and the second time may be interpreted as being closer to Baddie Flattery.
In Rurouni Kenshin, during his time as The Dragon to Takeda Kanryu, Aoshi Shinomori has just unleashed his illusionary sword technique to inflict a (he thinks) fatal wound to Kenshin. Upset at this, Yahiko declares that he's not going to run away even if Aoshi kills him. Prompting this response from Aoshi (and fortunately Kenshin gets back up shortly afterward):
Aoshi: You've got spirit. Too bad you're going to die here...
A later example is seen when Hajime Saito first appears, arriving at the Kamiya Dojo while posing as a medicine man. Sanosuke immediately sees through his disguise and punches him in the face (in the manga, hard enough to send him flying). However, while Saito suffers absolutely no ill-effects from the punch, he commends Sano's strength even as he then outlines why the punch was ineffective.
Saito: Your punch is worthy of your reputation. However...your punch was born during the peaceful Meiji era. It would not have any effect on a fighter from the Bakumatsu.
When Raoh first fully appears in the series, he holds Toki, one of Kenshiro's staunchest fighting allies, in considerably high regard--in fact, for him, Toki is the only fighter around who could conceivably match him in a fair duel, the only drawback being that Toki is suffering from a terminal illness caused by nuclear fallout. It helps that Toki was previously seen as The Messiah and, prior to the nuclear war, was considered (including by Kenshiro himself) to be the most suitable candidate for Hokuto Shinken successor and would probably have gotten it were it not for his illness. (It also says something when even Jagi is acknowledged by Ken to have admired Toki greatly.)
Raoh also gives this to Ken himself following their final fight (initially, Raoh had been dismissive of Ken being the Hokuto Shinken successor). It pulls multi-duty as a Final First Hug as well as a heartwarming and sad moment.
Raoh: Come, let me see the face of the man who has defeated Raoh...You are magnificent, my little brother.
Kenshiro: Big brother...
In Naruto, Itachi's interactions with the titular character have been quite polite for an antagonist, and he ends up giving a smile when Naruto declares that he will bring back Sasuke to Konoha. As well, he acknowledges Maito Guy's Crouching Moron, Hidden Badass credentials, escaping as soon as the latter shows up when he and Kisame first show up in the series. This may be a complicated example, though, as he was never truly a bad guy to begin with.
Noah: You possess both the skill and judgement necessary to catch your enemies off guard the second they lose focus. When you think about it, you and I have no reason to be enemies. If you swear your loyalty to me, I'll save all your friends from hell this very instant!
This basically defines how Lupin, the titular Villain Antagonist of Lupin III, feels about Inspector Zenigata. After all, Zenigata's dedicated and determined enough to keep on Lupin's trail no matter how many times the master thief dodges him, but is also reasonable enough to realize that many times the criminals they both have to face off against are far worse than Lupin himself. As a result, Lupin will go out of his way to ensure Zenigata isn't killed, while Zenigata in turn will sometimes allow his rival to escape just so the chase can continue.
This characterizes Ra's al Ghul's attitude toward Batman, as he consistently addresses the Dark Knight as "Detective" as a show of respect to the hero's intelligence and determination.
In DC's Richard Dragon series, Richard battles against his Rival and Arch-Enemy, Lady Shiva (also a Bat-familyAnti-Villain). Both of them have been, at different points in history, considered the greatest martial artists in the world, but Lady Shiva believes that the only true fight is a one-on-one duel to the death. During their match, Richard gains a decisive advantage and is poised to deliver a fatal attack, the Leopard Blow. With Shiva sure to die, one of her over-protective Mooks jumps in to save her. An angry Shiva immediately kills the Mook and apologizes to Richard for the interruption. They resume the fight, and this time Shiva comes out on top, killing Richard. She expresses sincere regret that the fight ended this way, because by all rights Richard should have won.
Magneto has this attitude toward Charles Xavier, despite their widely-differing views on human-mutant relations (it helps that he realizes they both want what's best for mutantkind, even if their methods can't mesh).
This is emphasized in the third live-action film, when he chews Pyro out for indicating he (Pyro) would have assassinated Charles Xavier if Magneto had given the word.
In The Flash, the titular speedster's Rogues Gallery tends to have this as their defining trait toward their nemesis (prominently toward Barry Allen when he was wearing the mask during the Silver Age, but also toward Wally West as well), to the point that their code of conduct forbids them to ever kill any Flash. It helps that most of them have standards to begin with. (However, this doesn't carry over to their Justice League Unlimited incarnations, where they're more than willing to kill Wally West at the first opportunity.)
It does carry over to their incarnations in Batman: The Brave and the Bold, albeit limited to Barry Allen (they are dismissive of Jay Garrick and Wally West).
In Deryni Rising, Charissa finds Morgan in the royal library while he researches Kelson's empowering ritual. During their tense verbal fencing match, she mentions having just looked in on the prince as he slept, and Morgan insists his magical Wards Major are impenetrable.
Charissa: That is probably true. You set your wards most effectively. In fact, even I was impressed with your skill. I had thought a half-breed Deryni incapable of such highly developed expertise.
Voldemort to Neville (former Butt Monkey) at the end of Harry Potter and the Deathly Hallows: "You show spirit, and bravery, and you come of noble stock. You will make a very valuable Death Eater. We need your kind, Neville Longbottom."
The Bible gives us an example in Nebuchadnezzar, king of Babylon. Although he overthrew Judah, as documented in Jeremiah 39, he gave orders that the prophet Jeremiah should be well taken care of (it helps that Jeremiah, under orders from God, had tried to convince the Jews to submit meekly to Nebuchadnezzar instead of trying to revolt). Later on, in the book of Daniel, Nebuchadnezzar showed regard to Daniel as the man who correctly interpreted his dreams (despite throwing Daniel's three friends into the fiery furnace). Both cases occur before his Heel-Face Turn, which he himself talks about in Daniel 4.
Live Action TV
In Babylon 5 episode "The Illusion of Truth," commentors on the martial-law-era ISN network make a point to praise Sheridan as a great hero of the Earth-Minbari War, but put on "experts" who state that he was psychologically damaged and fell prey to "alien influences" through no fault of his own.
Dhaos from Tales of Phantasia praises the heroes for saving the World Tree from its death, something he wanted to avoid in the first place. This doesn't stop him from engaging them in a Final Boss fight seconds later.
Vega:(to Chun Li) Your beauty and strength impress. I shall remember this day.
Juri:(to Ryu) I enjoy taking on fighters like you. You actually put up a fight.
Seth:(generic post-fight quote) I did not expect you to put up such a fight. You have impressed me.
In Assassin's Creed III, after Connor kills his father Haythem, before he dies he tells Connor that he's proud of the qualities Connor has (having great conviction, strength, and courage).
In Dragon Age II, you can earn the respect of the Qunari Arishok (the Final Boss of Act II) by always staying honest and blunt with him and his people. Doing so even earns you the achievement "A Worthy Rival".
Five hats means that five tropers think it is ready to publish.
You are saying that you think this draft is ready to be published. That means the description is not ambiguous,
it doesn't duplicate an existing trope, there are at least three examples, and the title makes sense.
Is that what you meant to do?
You are saying this draft has a ready-to-publish hat it does not deserve and you are taking it back.