I've recently been studying group cohomology, the original definition I learned was that of Ext, where $H^n\left(G, M\right)= \text{Ext}_{\mathbb{Z}G}^i\left(\mathbb{Z}, M\right)$. I then read a source which used the original definition of cohomology, as given in https://www.jstor.org/stable/1969215?seq=3#metadata_info_tab_contents. Where we consider the cochain groups $C^n\left(G, M\right)$ consist of functions $F:G^{n+1} \to M$ satisfying the homogeneity condition that $F\left(xx_0, \ldots, xx_n\right)=xF\left(x_0, \ldots, x_n\right)$.
How can we recover the original definition from the Ext one? I know that we can take any projective resolution of $\mathbb{Z}$ over $\mathbb{Z}G$ for example:
$$\ldots \to \mathbb{Z}G^2 \to \mathbb{Z}G \to \mathbb{Z} \to 0 $$ We can then calculate the Ext groups via taking the cohomology of
$$0 \to \text{Hom}_G\left(\mathbb{Z}G, M\right) \to \text{Hom}_G\left(\mathbb{Z}G^2, M\right) \to \ldots $$
I know now that we have an adjunction
$$ \text{Hom}_{Rings}\left(\mathbb{Z}G, R \right) \cong \text{Hom}_{Groups}\left(G, R^*\right) $$ But this doesn't seem to be enough to recover the original definition. Is there some other trick to do this?