Let $K$ be a field in $A$. If $a_1,\dots, a_n\in A$ prove there exists a unique ring morphism $\phi:K[x_1,\dots, x_n]\to A$ such that $\phi(x_i)=a_i$

Question

Let $K$ be a field in $A$. If $a_1,\dots, a_n\in A$, prove there exists a unique ring morphism $\phi:K[x_1,\dots, x_n]\to A$, such that $\phi(x_i)=a_i$

I'm actually not entirely sure I understand the question. From my understanding since $K\in A$ and $K[x_1,\dots, x_n]$ has coefficients in $K$, I can define a ring morphism $\psi:K[x_1,\dots, x_n]\to K$ such that for $k_1,...k_n \in K$, $\psi(x_i)=k_i$ and then define $f:K\to A$ as $f(k_i)=a_i$

Then for $f\circ\psi$, I prove that this is a ring morphism. But would the uniqueness part be proving that $\phi=f\circ\psi$ or proving something else?

Answer 1

Question answered in comments.