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What about Sour Supporter?
It often happens that, maybe one member of a pair refuses to let his or her partner go on a mission, but the partner says that it must be done, and he or she must go. "Must you go?" "I must." "Then, I shall go with you." Emotional moment ensues. I don't think this person is quite "Sour"... He or she is probably better described as "Selfless Skeptic".
This seems like different scenarios. (a) The hero not wanting the supporter to come along. (b) The supporter refusing to come along but changing his mind quickly. There is also Hero Harasses Helpers.
Edited by eroock
This happens often in shounen (shonen) anime, but also happens in western TV dramas and movies (I'm thinking maybe Fast and Furious). One character in the protagonist group is initially against the idea of going on a particularly risky mission, but it's the only way (or perhaps not). In the last moments, he or she declares that if everyone is set on going then, "I'm going with you!"