= Solution
If $N_G(D)\le H\le G$, then $N_H(D)=N_G(D)=N$ and $C_H(D)=C_G(D)=C$. Applying part (d) to $G$ and to $H$ therefore gives the same target <ideal>
$$
J_D=\operatorname{Tr}_D^N(kC).
$$
Compose the two bijections. A block <idempotent> $e$ of $G$ with defect $D$ corresponds to the unique block <idempotent> $f$ of $H$ with defect $D$ satisfying
$$
\boxed{\operatorname{Br}_D^G(e)=\operatorname{Br}_D^H(f).}
$$
This proves the requested form of <Brauer first main theorem>, including uniqueness and the equality of the Brauer images.
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