Describe all ring homomorphisms from $\mathbb{R}[T] \rightarrow \mathbb{R}[T]$

The only Homomorphism on $\mathbb R$ that fixes $1$, is the identity map. For the proof you can see Ring homomorphisms $\mathbb R\to\mathbb R$ .
Since $f$ is a homomorphism that fixes $1$, image of every invertible elements in $\mathbb R[T]$ should be invertible, and we should have $f_{|\mathbb R}=Id_{\mathbb R}$ .

$f$ completely determined by the image of $T$, and so

$f(a_0+a_1T+...+a_nT)=a_0+a_1(p(T))+...+a_n(p(T))^n$ where $p(T)=f(T)$.

At last $f$ is isomorphism iff $f(T)=aT+b$