I believe this is how we define Cor map: Suppose we have $G$ module $M$ and finite index subgroup of $G$, say $H$. Then Milne says (in his Notes on CFT) that we have, by Shapiro's lemma, $$H^r(H,M) \xrightarrow{\cong} H^r(G, Ind_H^G(M)) $$ given by map induced by inclusion of $H$ in $G$ and $\phi \mapsto \phi(1_G) $.We do not prove that this map is isomorphism but simply state that this is the map from the proof of Shapiro's lemma.(https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.2.1.27).
But Shapiro's lemma (https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.2.1.11) was proved without using any explicit map so how do I know that the map I have given above is indeed the map in Shapiro's lemma. One way, obviously, is to keep track of all the maps used in proving Shapiro's lemma. This, I believe, could be done in principle but seems rather painful and I think there must be a better way around.
Then we define $Ind^G_H(M)\rightarrow M $ as $\phi \mapsto \sum_{s \in S} s \phi (s^{-1}) $ where $S$ is set of left coset representatives of $H$ in $G$. This induces a map $$H^r(G, Ind^G_H(M)) \rightarrow H^r(G,M) $$
Composing this with above map from Shapiro's lemma we get the corestriction map.
What is our purpose behind this definition i.e. what is motivation for corestriction map?
Any help is appreciated and feel free to give any reference.
To show that the two maps agree, check that they agree for $r=0$ and then use dimension shifting (see II 1.28 of the notes). Specifically, use induction on $r$. Choose an embedding $M\to I$ of $M$ into an injective module, and let $N=M/I$. The two maps agree on $N$ for $r-1$ (induction hypothesis) and it follows that they agree on $M$ for $r$.
What's the motivation for the corestriction map? Well, its heavily used in the notes, for example to show that $H^r(G.M)$ is killed by the order of $G$.