Why is Mycroft willing to help Sherlock in "The Hounds of Baskerville" case?
We already know about their "sibling rivalry". And for me, the BBC series depicts their rivalry more intensely than the original by Sir Arthur Conan Doyle. In other words, BBC's Mycroft loves to "make-it-hard" for Sherlock just for the sake of "rivalry".
Coming back to the question. The Hounds of Baskerville isn't a matter of national security, so I guess it wasn't really important for Mycroft to be involved. While there is nothing Mycroft can gain from that case, we see Mycroft grants Sherlock his request for full access to the Baskerville army base on his second visit.
Why would Mycroft do that?
Well, I disagree with the fact that their rivalry is shown in a more intense manner by BBC. They don't wish bad for each other rather they are dependent on one another.
Mycroft asks for Sherlock's help when there is legwork involved and Sherlock on the other hand, needs Mycroft's help for granting him official access/government authentication. Both require each others' assistance at different places. There is no solid reason for Mycroft to refuse Sherlock the special grant.
Their relation in the TV series is graduating every season, From a sibling feud to a true brotherly bond.
In this season their relation is a bit unexplored. It suggestive of a sibling feud owing to a common past.
Just when Mycroft comes to know about John, being a potential flatmate of Sherlock, he stalks him and even asks to spy on him.
Mycroft asks Sherlock to help him fetch the missile plans.
Here, the relation is seen more as a sort of dependency. It is a give and take relationship.
Sherlock helps Mycroft on the Irene Adler case.
Mycroft grants Sherlock full access in Baskerville army base on his 2nd visit.
This season shows that Sherlock looks up to Mycroft as a proper big brother and for Mycroft, Sherlock is his younger sibling.
At John's Wedding the courtroom scene clearly showed that Sherlock had head Mycroft and looks up to him for advice when he is stuck.
When Sherlock shoots Magnussen, through Mycroft's point of view we see a young boy crying, suggestive of the fact that for Mycroft, Sherlock will always be a small brother.
Their relation matures up in this season.
On learning that Sherlock had to shoot either John or him, he starts being obnoxious just to make it easy for Sherlock to shoot him
Towards the end, Sherlock asks Lestrade to take good care of Mycroft.
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