Linear structure on the category of formal groups

Question

Let $R$ be a commutative ring. If $R$ is a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ (or the category of formal group laws) carries the structure of an $R$-linear category; this is because it is equivalent to the category of Lie algebras over $R$ (which are free of finite rank as $R$-modules) by the $\mathbb{Q}$-Theorem, and the latter category is $R$-linear. If $R$ is not a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ doesn't have to be $R$-linear. Nevertheless, I would like to ask: Is there any way to see the $R$-linear structure on the category of formal groups over $R$ directly, without the classification in terms of Lie algebras, when $R$ is a $\mathbb{Q}$-algebra? For example, why should be expect there to be a map $R \to \mathrm{End}_R(G)$ for every formal group $G$ over $R$?

Answer 1

If $G$ is a formal group over $R$ and $R$ is a $\mathbb Q$-algebra, its logarithm is defined over $R$ (see the explicit formula for the logarithm of a formal group) thus giving an isomorphism from $G$ to the additive formal group $F_a$. It remains to note that ${\rm End}_R(F_a)$ is isomorphic to $R$.