 Green correspondence from Mackey multiplicity

Let $N_G(D)\le H\le G$ and let $U$ be indecomposable with vertex $D$ over $H$. The Mackey restriction formula writes ${\mathrm{Res}}_H^G{\mathrm{Ind}}_H^{G}U=U\oplus T$, where the vertices in $T$ lie in $H\cap {}^{g}D$ for $g\notin H$. None of these subgroups contains an $H$-conjugate of $D$: equality would put $g$ in $N_G(D)H=H$. Every summand of the induction with a $G$-conjugate of vertex $D$ contributes a vertex-$D$ summand on restriction. Krull-Schmidt decomposition therefore forces exactly one such summand, with multiplicity one. Choosing sources and using transitivity of induction proves the inverse correspondence and preservation of sources.