= Solution
A Lubin–Tate series for $\pi$ is a power series $f(X)\in\mathcal O_K[[X]]$ such that
$$
f(X)\equiv\pi X\pmod{X^2},
\qquad
f(X)\equiv X^q\pmod\pi.
$$
The key Lubin–Tate lemma is that if $f$ and $g$ are such series and $L(X_1,\ldots,X_r)=\sum_i a_iX_i$ with $a_i\in\mathcal O_K$, there is a unique series $F\in\mathcal O_K[[X_1,\ldots,X_r]]$ satisfying
$$
F\equiv L\pmod{(X_1,\ldots,X_r)^2},
\qquad
F(f(X_1),\ldots,f(X_r))=g(F(X_1,\ldots,X_r)).
$$
Taking $g=f$ and $L=X+Y$ defines $\mathcal F_f(X,Y)$. Uniqueness applied to the two sides of each identity proves the identity, associativity, and commutativity axioms, so $\mathcal F_f$ is a <formal group law>. Taking $L=aX$ defines endomorphisms $[a]_f$, and uniqueness gives
$$
[a+b]_f=\mathcal F_f([a]_f,[b]_f),
\qquad
[ab]_f=[a]_f\circ[b]_f.
$$
Thus $\mathcal F_f$ is the <Lubin–Tate formal group> as a formal $\mathcal O_K$-module, with $[\pi]_f=f$.
For another Lubin–Tate series $g$, apply the lemma with $L=X$ to obtain
$$
h(X)\equiv X\pmod{X^2},
\qquad h\circ f=g\circ h.
$$
Uniqueness shows that $h$ respects both formal addition and every scalar endomorphism. Its linear coefficient is the unit one, so it has a compositional inverse; equivalently, applying the lemma with $f$ and $g$ reversed supplies the inverse. Hence this <Lubin–Tate change of series> is an isomorphism $\mathcal F_f\cong\mathcal F_g$ of formal $\mathcal O_K$-modules.
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