I met a question in group cohomology while reading Milne's $Class\space Field \space Theory$
Now the $Shapiro's \space Lemma$ gives an isomorphism $H^r(G,Ind^G_H(N)) \to H^r(H,N)$, this isomorphism can be given by the composite
$H^r(G,Ind^G_H(N))=H^r({Ind^G_H(I^\bullet)}^G) \to H^r({I^\bullet}^H)= H^r(H,N)$
For any subgroup $H$ of $G$, and every $G$ module $M$, we have the restriction homomorohism
$Res:H^r(G,M)\to H^r(H,M)$
It is said in this book that this restriction map is just the composite of the following two maps:
$H^r(G,M)\to H^r(G,Ind^G_H(M))\simeq H^r(H,M)$
Where the first map is induced by the homomorphism
$M\to Ind^G_H(M):m\to(\varphi_m:g\to gm)$
My question is how to verify that this composition is the same as the restriction map $Res$