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In Big Hero 6, one of the central plot points is when Hiro tries using Baymax to kill Robert Callaghan. He tries to do this to avenge his brother, who died in a fire at the university Callaghan taught at.
Is it ever stated explicitly that Callaghan started that fire? It is certainly implied that way, but I don't recall Callaghan straight-out admitting it.
In the tie-in Junior novels, it's stated that Hiro worked out that Yokai/Callaghan started the fire to cover up his crime, but he bases this on no evidence other than learning of the theft.
Hiro was stunned. Callaghan had started the fire that killed Tadashi!
and
Then he [Hiro] realized something: “The guy in the mask! He set the fire so no one would know he had stolen my microbots!”
Similarly, the Big Hero 6 Essential Guide states that Yokai (Callaghan) must have started the fire, but again, this is based on Hiro's assumption.
This sinister super villain hides his identity behind a scary mask. Hiro realises that Yokai must have started the fire at the Tech Showcase and stolen his microbots, but why?
We also have some light confirmation from the children's reader book Big Hero 6: The Big Battle that Yokai intentionally caused the fire.
There is a fire at the showcase! Someone set the fire to steal the microbots.
The Art of Big Hero 6 states that Tadashi died in a tragic accident. Clearly Professor Callaghan intended for the fire to act as a diversion, allowing the theft of the microbots and his personal disappearance, but didn't expect there to be any human casualties.
Big Hero 6 is the fifty-fourth film made by the Walt Disney Animation Studios. But it's their first bona fide super hero action movie. The story follows Hiro Hamada, a teenage genius, who is left adrift when his beloved older brother, Tadashi, dies in a tragic accident. Together with his robot, Baymax, Hiro leads a reluctant team of first-time super-heroes in a fight to stop a masked villain from destroying their high-tech, fast-paced city, San Fransokyo.
In Big Hero 6, Callaghan never discusses the fire except right after he is revealed to not only have survived the fire, but ...
to be the villain of the movie.
At this point, the only dialog is:
Hiro: Pro ... Professor Callaghan. The fire ... You died.
Callaghan: No. I had your microbots.
Brief interlude showing Callaghan putting on the transmitter and forming a microbot shell around himself.
Hiro: But, Tadashi. You just let him die.
Callaghan: Give me the mask, Hiro.
Hiro: He went in there to save you!
Callaghan: That was his mistake!
After this, Hiro orders Baymax to destroy Callaghan.
However, earlier in the movie, Hiro theorizes that the "guy in the mask" stole his microbots and set the fire to cover his tracks. While Callaghan does not confirm this, this is the only explanation we have for the fire in Big Hero 6.
Additionally, Disney's cast announcement does imply that Callaghan (called Yokai as the villain) started the fire. If that is the case, than Hiro's theory is most likely correct.
YOKAI is the silent masked man behind the terrible tragedy that strikes San Fransokyo, turning Hiro’s world upside down. As a result, Hiro transforms a group of nerds into a team of high-tech crime fighters—Big Hero 6—with one mission: Track down Yokai and bring him to justice.
Finally, Big Hero 6: The Series picks up after the events of the movie. As of October 2018, Callaghan has only appeared in the episode "Mini-Max," where he is remorseful for what he did in the movie. However, the only explanation he gives is:
Callaghan: I never set out to hurt anyone. What happened to Tadashi, because of me, I know it's not enough, but I'm sorry.
Callaghan is never stated to have started the fire in the Big Hero 6 movie or TV show, so we cannot prove that he did. However, the cast announcement and Hiro's hypothesis imply that he may have done so, or at least did not rescue Tadashi when he had the chance. Because the connection could have been further explained and hasn't, I would assume that Disney's intention is to keep it ambiguous.
