Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.

Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{Gal}(L_w/K_v), E(L_{v_L})), $$ where $\operatorname{Gal}(L_w/K_v)$ denotes the local Galois group corresponding to the place $v$.

For an abelian group $M$, let $M^*$ denote its Pontryagin dual.

In this context, the Tate-cohomology group is represented as $$ \hat{H}^0(\operatorname{Gal}(L/K),E(L)) = E(K)/(1+\sigma)E(L). $$(In other words, if we define a map $trace:E(L)\to E(K), P\to P+P^{\sigma}$, it is $ Coker(trace)$.)

According to p214 of the cited paper (link), it's claimed that $$ {cokerF}^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $$ but this is asserted without further explanation.

Could you provide some insights or suggest strategies for proving this isomorphism?

N.B. The linked paper reads the dual of diagram $(3)$ in p214 is the bottom diagram in p214. lem5 in p214 follows immediately if we confirm the titled isomorphism.