The enemies.

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The enemies.
In 1241, at the height of the Mongol Conquests, the son and successor of Chinggis Khan, Ogedai Khaan, died. His wife, Torogene Khatun, was left as regent of the great Empire, which at that time stretched from Kiev to Korea and from Moscow to Iran. This Woman for 5 years ruled the largest and most powerful empire on the planet, which would in the years following go on the conquer the rest of China and much of the Middle East. She secured the Mongol throne for her son Guyuk and was by far one of the most powerful women in history.
Guyuk khan 🔥
Mongolian princes in Waste conquest arguing about tactics.
Batu & Buri
Mengu & Subutai Baatar & Guyuk
Guyuk is trying to get to Karakorum earlier than Batu's envoy with message for kaan.
Four young princes in Chinese conquest.
Batu and Mengu.
Guyuk kaan. Son of Ogedei.
On the antagonism between Guyuk and Batu
A response to @annedey repeated here in full so it may be easier to read my answer. I don’t understand Tumblr’s formatting when it comes to responses.
The Batu-Guyuk antagonism is one of those things which scholars always note, but do not tend to actually go into anymore than that. McLynn uses the extensively annotated Secret History of the Mongols by Igor de Rachewiltz, and while I don’t have access to de Rachewiltz’s I thumbed through other sources to see if there was more I could find and I found… not much more.
But I believe I was able to gather some reasons as to why they disliked each other:
Timothy May suggests this had to do with Jochi’s uncertain paternity. Batu, while not Jochi’s eldest son was his most prominent and the chief prince of his Ulus. Thus he ‘inherited’ Jochi’s uncertain lineage, which has been suggested kept Batu and his brothers from ever putting their names in to be Great Khan. The fact that none the less Batu maintained such a high status seemed to aggravate Guyuk. The Secret History of the Mongols, in which this confrontation appears in (where they call each other women etc.) has Buri a major player in this, a close friend of Guyuk. Buri’s grandfather of course was Chagatai, who was famous for his hatred of Jochi, a hatred which got them both removed from the succession to Chinggis. I bet good money that was certainly the reason why Buri disliked Batu, and if Guyuk was a close associate of Buri this was probably a leading factor as well.
Princely arrogance was another strong factor. All of them were princes of an Empire seemed destined to rule the world, already a recipe for a lack of humility. Guyuk, as noted by Rashid al-Din, was an exceptionally arrogant individual even among them, augmented by the fact that his father the Great Khan of this empire (although this did not mean he was the heir of Ogedai: Ogedai’s desired heir was his grandson Shiremun, but the Mongol succession system did not really have ‘heir apparents’ in the same way western Europe did). But Batu was considered the leading prince of the expedition, and the territory they were conquering would certainly go to Batu’s ulus, which must raised the ire of Guyuk: he was essentially fighting to gain territory for a person who he may not have even considered a legitimate Chinggisid.
Past that, the exact origins of the antagonism are unclear. If we look at the Secret History of the Mongols, the implication is that they had often butted heads, likely over minor things, with the occasion McLynn discusses being that Batu began drinking before the others at a feast, thereby signifying his status over the rest (or rather, Guyuk and Buri were taking offence at small little things to make a fuss out of them). Buri was apparently notorious hothead and drunk (big surprise, exactly like Chagatai) which I imagine started off more than few rows between them.
There have been other ‘attempts’ to explain their distaste for each other: Conn Iggulden, who is an author and not an historian, has Guyuk possibly homosexual in his novels and uses that as an explanation for break between them, but I have seen no historical evidence to suggest this.
In the end, I’d probably say the lineage and arrogance of both parties was the reason for their hatred, but as far as I know there is no official “x is why they hate each other.”