# Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$

Prove: Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$.

Note: In my book it defines $E_g\colon K[x] \to K[x]$ by sending $x$ to $g$.

This seems like it just follows from definitions but I guess I am just looking for some reassurance in my thought.

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Hint: the map is already additive, and is multiplicative on powers of $x$, and fixing elements of $K$ amounts to it being linear. If $\phi(x)=g\in K[x]$, verify that the mentioned properties determine the image of an arbitrary polynomial.
P.S.: The notation $E_g$ is supposed to suggest "evaluation".