I am reading Milne's CFT notes and am confused by his proof of Shapiro's lemma. Shapiro's lemma states that $H^r(G, Ind_H^G(N))\cong H^r(H, N)$. Milne's proof first shows that $N^H\cong Ind_H^G(N)^G$, which proves the $r=0$ case. Then he takes an injective resolution $N\rightarrow I^\bullet$ and applies the exact functor $Ind_H^G$ to get an injective resolution for $Ind_H^G(N)$.
I follow all this so far. Then he claims that $H^r(Ind_H^G(I^\bullet)^G)\cong H^r((I^\bullet)^H)$, using the $r=0$ case as justification. Unfortunately I am unable to see why the isomorphism at the 0th level extends to an isomorphism of all cohomology groups. I feel like it may have something to do with the invariance of cohomology wrt different injective resolutions, but the fact that these complexes come from different functors (one fixed by $H$, the other fixed by $G$) make me confused as to how to relate their cohomologies.