Everyone is sharing their songs for Master Kohga, so I have to hop in >:J
Let me put you on Backlight by Ado!
This song really captures my impression of both Kohga's character and circumstance:
The Yiga Clan broke away from the Sheikah thousands of years ago after the betrayal by the Royal family, stirring a lasting resentment in the people that persisted until Kohga's reign. This betrayal was deeply damaging to both the Sheikah and Yiga of the time; however, the Sheikah remained loyal to the throne despite the callous disregard for them. The Yiga found this compliance to be galling and detrimental to their entire people as they allowed this subjugation.
Due to this, the Yiga wanted to punish the Sheikah for their "love" and continued loyalty to the throne that so easily took from their people, effectively discarding them; however, the Yiga possess that same "love" for Calamity Ganon, who would, and had discarded them once their use left them.
Kohga suffers from this conviction: he is physically pained by his stress*, and his people are killed not only by their foes, but by the one they've served for thousands of years.
Even through it, Kohga continues on his path; The anger and resentment of the Yiga became tradition, deeply engraining itself into their society and, more so, in Kohga himself. In how he was raised to become the future leader of the Yiga, he became deeply convinced of their people's path and ambition. Each death only solidified Kohga's conviction- he couldn't let his people die for nothing- for a delusion of purpose.
That was until Sooga died.
Sooga never truly held the clan's conviction; The second was born an outsider and saw through the illusions and false promises deeply ingrained into the Yiga conscience. In this way, he didn't really serve the clan and its goals, rather he served Master Kohga directly. He went along with it all as an extension of Kohga's desires, even as their home and people crumbled.
Only then could Kohga let go of this resentment, even if it was just to take on a new resentment. Unlike the Sheikah, the Yiga were subjugated twice- once of their own volition- and only after Sooga died, could Kohga see that.
Kohga is a deeply flawed man. To him, the depth of the injustice committed by Hyrule so many years ago justifies any violence, no matter the damage it inflicts on others, his own, and even himself. While his love for Sooga in AoC showed him that this hatred wasn't worth the price, cumulating in him changing sides, this lesson wasn't learned in the BoTW timeline.
With Dorian, it wasn't that Kohga was apathetic. Dorian, in leaving the Clan to rejoin the Sheikah, rejected the foundation of their people; To someone like Kohga, that was nothing short of personal betrayal, a somewhat unremarkable disloyalty internalized by thousands of years of hatred.
uhh, sorry if I sound delusional...This picture is how I feel right now T_T
*This is a headcanon from his "stress" mechanic in AoC.
