Saigou hailed from Satsuma, famous for its loyalists and its hatred of Choshu. Katsura hailed from Choshu, famous for its loyalists and its hatred of Satsuma. Sakamoto came along and realized that Choshu needed modern weapons, which Satsuma had due to a trading alliance with a Scotsman, and Satsuma needed more support for the sonno joui movement, which Choshu had in the form of hotheaded rebellious youth.
In the interest of getting the Shogunate out of power, Katsura and Saigou met through Sakamoto and worked out an alliance between the two domains called the Satcho Alliance. As a part of this agreement, Sakamoto was to use his new company to supply Choshu with weapons smuggled from Satsuma.
Katsura and Saigou continued to hate each other well after the collapse of the Shogunate and into the establishment of the new Meiji government, and kept using their newly appointed governmental positions to try and screw the other over. (Katsura eventually won, but that’s a lesson for another time.)
Katsura Kogoro, who by then had changed his name to Kido Koin or Kido Takayoshi, was on his deathbed during the midst of the Satsuma Rebellion, during which Saigou Takamori rebelled against policies Katsura’s compatriots put into place for the new Meiji government. Katsura’s last words came as troops from the newly founded Imperial Army were marching to Satsuma in order to put Saigou’s rebellion down.