Let $R_0$ be a subring of a commutative ring $R$, we denote by $\mathrm{Mod}_{R_0}$ (resp. $\mathrm{Mod}_R$) the category of modules over $R_0$ (resp. $R$). We have the extension of scalars functor $\mathcal{F}:D\in\mathrm{Mod}_{R_0}\mapsto D\otimes_{R_0}R\in\mathrm{Mod}_R$ w.r.t the embedding $R_0\rightarrow R$. Its right adjoint functor $\mathcal{G}:\mathrm{Mod}_R\rightarrow\mathrm{Mod}_{R_0}$ is the restriction of scalars.
Now let $\mathcal{C}$ be a subcategory of $\mathrm{Mod}_{R_0}$. If $\mathcal{F}\mid_{\mathcal{C}}$ induces an equivalence between $C$ and a subcategory $\mathcal{D}$ of $\mathrm{Mod}_R$, is $\mathcal{G}\mid_{\mathcal{D}}$ a quasi-inverse of $\mathcal{F}\mid_{\mathcal{C}}$?
If this is not always the case, what conditions (on $R_0$, $R$, $\mathcal{C}$, $\mathcal{D}$) can be added to make it work?