Let $R$ be a commutative ring and let $F=F(R,R)$ be the set of functions $f:R\rightarrow R$. Let $I$ denote the identity function $I(r)=r$ in $F$. Prove that the map $\varphi:R[X]\rightarrow F$ given by $$\varphi (a_nX^n+\cdots+a_1X+a_0)=a_nI^n+\cdots+a_1I+a_0$$ is a ring homomorphism.
I easily showed that $\varphi$ is a group homomorphism from $(R[X],+)$ to $(F,+)$. Now I'm stuck on how to show $\varphi(xy)=\varphi(x)\varphi(y)$ for all $x,y\in R[X]$. How should I tackle this proof?
I want to show that $$\varphi[(a_nX^n+\cdots+a_1X+a_0)(b_mX^m+\cdots+b_1X+b_0)]=\varphi(a_nX^n+\cdots+a_1X+a_0)\varphi(b_mX^m+\cdots+b_1X+b_0)$$ but I don't know whether I should combine the terms through multiplication because it's hard to determine what happens inside the dots $\cdots$. Should I expand or is there a better way of doing this?