On $A$ algebra homomorphisms $A[[X_1,...,X_n]]\to Q(A)$, where $A$ is a complete DVR

Let $\pi$ be a uniformizer of $A$. Suppose $\phi(X_{i}) = u/\pi^{m}$ for some $u \in A^{\times}$ and $m \ge 0$. Then $\pi^{m}X_{i}-u$ is a unit of $A[[X_{1},\dotsc,X_{n}]]$ that gets mapped to $0$ in $K$, contradiction.