Answer based on context:

The Tolui family successfully enthroned their candidate for Great Khan, Möngke, in the kurultais of 1250 and 1251.  The Ögedeid candidate for Great Khan, Shiremun, and his cousin Nakhu, were embittered by their loss and plotted a failed assassination of Möngke.  Möngke took revenge by purging his opponents in the royal house, and members of both the Chagatai and Ögedei families. Möngke handed control over the Caucasus region to the Golden Horde in 1252.  With the approval of Möngke, Berke succeeded his brother, Batu, as Khan of the Golden Horde in Russia in 1255.  Hulagu of the Ilkhanate seized control of the Caucasus from the Golden Horde, and his sacking of Baghdad  in 1258 angered Berke, a convert to Islam.  Möngke Khan died in 1259 without appointing a successor. He likely favored Ariq Böke, whom Möngke designated in 1258 as commander of Karakorum , but he did little else to secure Ariq Böke's claim to the throne.

Who were the two candidates for Great Khan?
Möngke