Let $k$ be an arbitrary field and consider the corresponding one point field scheme $Spec(K)$. Set $G:=\text{Gal}(K_s/K)$ the corresponding Galois group of separable closure.
Denote by $\mathbf{Sh}\big(\text{Spec }k)_{\text{et}}$ the category of etale sheaves over $Spec(k)$.
I often heard that known theorem that the exist a category equivalence between
$$ A: \mathbf{Sh}\big(\text{Spec }K)_{\text{et}} \cong\{\text{category of discrete G-modules}\} $$
given explicitely by
$$\mathcal F \mapsto \varinjlim_{K'/K\text{ finite separable extension}} \mathcal F(\text{Spec }K') $$
with inverse $$\big(\text{Spec }k'\mapsto M^{\text{Gal}(\overline K/K')}\big) \leftarrow M $$
My question question is why this category equivalence also induces the identification of Etale cohomology with Galois cohomology?
In other words why $H^i_{et}(X,\mathcal F)\cong H^i(G,A(\mathcal F))$ resp $H^i_{et}(X,A^{-1}(M))\cong H^i(G,M)$ always holds for etale sheaf $\mathcal F$ and $G$-module $M$?