= Solution
A one-dimensional commutative <formal group law> over $R$ is a power series $F(X,Y)\in R[[X,Y]]$ satisfying
$$
F(X,0)=X,qquad F(X,Y)=F(Y,X),qquad
F(F(X,Y),Z)=F(X,F(Y,Z)).
$$
A homomorphism $\theta:\mathcal F\to\mathcal G$ is a series $\theta(T)\in TR[[T]]$ such that
$$
\theta(F(X,Y))=G(\theta(X),\theta(Y)).
$$
Write $\theta(T)=uT+O(T^2)$. If $u=\theta'(0)$ is a unit, recursive comparison of coefficients constructs a unique compositional inverse $\psi(T)\in TR[[T]]$ with $\psi(\theta(T))=T=\theta(\psi(T))$. Apply $\psi$ to the homomorphism identity and substitute $X=\psi(U)$, $Y=\psi(V)$ to obtain
$$
F(\psi(U),\psi(V))=\psi(G(U,V)).
$$
Thus $\psi$ is a homomorphism from $\mathcal G$ to $\mathcal F$, so $\theta$ is an isomorphism.
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