Lets $G=\hat{\mathbb{Z}}$ be the profinite completion of the integers, let $T$ be the topological generator of $G$. I'm interested in proving $$H^i(G,M)=\begin{cases} M^G & i= 0\\ M/(T-1)M &i=1 \\ 0 & i \geq 2 \end{cases}$$ for discrete torsion $G$-modules $M$. The usual approach is to use that profinite cohomology behaves well with limits and that compute the colimit of the cohomologies $H^i( \mathbb{Z}/n\mathbb{Z},M^{n\hat{\mathbb{Z}}})$.
I want to prove this statement using universal $\delta$-functors. My approach is to consider the category $Mod_t(G)$, the full subcategory of discrete torsion $G$-modules, it's an abelian category with enough injectives and the profinite cohomology on this category coincides with $H^i(G,-)$. Now let's consider the $\delta$-functor $F:Mod_t(G)\to Ab$, where we take $M \in Mod_t(G)$ and take the cohomology of the complex $M\overset{T-1}{\to} M \to 0 \to 0 \to \cdots$. Since $F^0(-)=(-)^G=H^0(G,-)$ it is enough to prove that $F$ is universal.
My obvious approach was to try to show that $F$ is effaceable functor, so actually I only need to show that $F^1$ is effaceable. This is where I'm stuck, since I have to pick something with non trivial $G$-action or else $M/(T-1)M=M$. Usually we pick the induced module or injective modules but I don't see why it's cohomology should vanish, since I should use somewhere that $M$ is torsion. I also thought about first reducing to the case where $M$ is finite, since every $M$ is a colimit of finite submodules.
I would appreciate any hints/ideas.