Suppose $K$ is a field and $A$ is a $K$-algebra. Let $End_K (K [x],A)$ denote the algebra of all endomorphisms from $K [x]$ into $A$. Then by the help of universal property it is very easy to see that there is a one-to-one correspondence $\psi$ between $End_K (K [x],A)$ and $A$ which sends $\phi_a$ to $a$ where for each $a \in A$, $\phi_a$ sends $x$ to $a$ and constants to themselves.But the domain and co-domain of $\psi$ are both algebras.So we may think further and can discuss whether it is an endomorphism or not. Then these two algebras are isomorphic to each other. But I think $\psi$ is not an isomorphism.Are these two algebras isomorphic to each other? If the answer is "yes" then what is this isomorphism?
Please help me in this regard. I am in a fix.
Thank you in advance.