This question comes from Silverman's Arithmetic of Elliptic Curves, specifically the appendix on Galois cohomology.
I am a cohomology beginner, interested (for now) in understanding just enough to get through a solid proof of Hilbert's Theorem 90: $$H^1(G_{\overline{K} / K}, \overline{K}^*) = 0.$$
I follow the proof in the case where $L / K$ is finite given in Serre's Local Fields, so the result would follow from an exercise in Silverman showing that if $M$ is a module on which $G_{\overline{K} / K}$ acts continuously (given the discrete topology on $M$ and profinite topology on $G_{\overline{K} / K}$, then $$H^1(G_{\overline{K} / K}, \overline{K}^*) \cong \varinjlim H^1(G_{L/K }, M^{G_{\overline{K}/ L}}).$$ Here, $M^{G_{\overline{K}/ L}}$ is the set of elements of $M$ fixed by $G_{\overline{K} / L}$. My background with direct limits is mostly from exercises in Atiyah Macdonald, so I would like to do this by showing that $H^1(G_{\overline{K} / K}, \overline{K}^*)$ has the universal property of direct limits. I can construct the direct system using the inflation maps, but I am stuck on how to define $\varphi : H^1(G_{\overline{K} / K}, \overline{K}^*) \to N$, such that for all finite Galois extensions $L$ of $K$, $\varphi_L = \varphi \circ \iota_L$, where $\iota_L$ is the inflation map from $H^1(G_{L / K}, M^{G_{\overline{K}/L}})$ to $H^1(G_{\overline{K} / K}, M)$, and $\varphi_L : H^1(G_{L / K}, M^{G_{\overline{K}/L}}) \to N$ is given (such that all the $\varphi_L$ are compatible with the direct system). How can we define such a map?
P.S. I know that this result appears in a lot of standard sources, but usually with much more power/background in homological algebra than I am looking for. Weibel has this result for profinite groups, but the proof uses $\delta$-functors, for example. The closest I've seen to what I'm looking for is in David Harari's Galois Cohomology and Class Field Theory, but even there I had trouble translating the general situation he presents into the concrete function that I'm looking for.