To prove (a), we pick any projective $R$-model ${\mathscr{Y}}$ of $Y$. By [Lü93, Lemma 2.2], there is a blowing up $\pi\colon{{\mathscr{X}}}^{\prime}\to{\mathscr{Y}}\otimes_{R}K^{\circ}$ such that ${{\mathscr{X}}}^{\prime}$ dominates ${{\mathscr{X}}}$. Since $\pi$ is a projective morphism, ${{\mathscr{X}}}^{\prime}$ is a projective $K^{\circ}$-model dominating ${{\mathscr{X}}}$. Hence (a) follows from (b).

To prove (b), we note that the morphism ${{\mathscr{X}}}\to{\mathscr{Y}}\otimes_{R}K^{\circ}$ is a blowing up morphism along a vertical closed subscheme $Z$ of ${\mathscr{Y}}\otimes_{R}K^{\circ}$ (see [Liu06, Thm. 8.1.24]). Since the ideal sheaf of $Z$ contains a power of the uniformizer of $R$, we may define it over $R$ and hence the same is true for the blowing up morphism and for ${{\mathscr{X}}}$ proving (b).

To prove (c), we may assume that the model function is associated to a vertical Cartier divisor $D$. Replacing $D$ by $D+{\operatorname{div}}(\lambda)$ for a suitable non-zero $\lambda\in R$ and using (a) and (b), we may assume that $D$ is an effective Cartier divisor on a projective $R$-model ${\mathscr{Y}}$ of $Y$. As in (b), we see that the ideal sheaf of $D$ is defined by the ideal sheaf of a Cartier divisor $D_{0}$ defined over $R$ proving (c). ∎