I am giving a presentation on the Brauer Group and the isomorphism $$H^2(K) = \underset{\longrightarrow_L}{\lim}H^2(G_{L|K},L^*) \cong \underset{\longrightarrow_L}{\lim} Br(L|K) = Br(K) $$ where $G_{L|K}$ denotes the Galois Group of a finite Galois extension $L|K$. However, we have not done anything on group cohomology and I am wondering why the inflation map $$inf_{F\subset L} : H^2(G_{L|K}/G_{L|F}, F^*) \to H^2(G_{L|K},L^*), \quad [\bar f] \mapsto [f]$$ with $$f: G_{L|K} \times G_{L|K} \to G_{L|K}/G_{L|F} \times G_{L|K}/G_{L|F} \overset{\bar f}{\longrightarrow}F^*\hookrightarrow L^*$$ used for the inductive limes of $H^2(K)$ yields an inductive system. Does anyone have good literature on this topic or a good explantation which does not require profound knowledge on group cohomology? In my script, it does not go in any details, but I want to understand deeply it in order to explain it.
Thanks in advance