Substitution principle example? (for ring homomorphisms $R[x]\to S$) 


I have to know the substitution principle for an upcoming exam. This is the definition given in my book. I understand the notation I believe. However, I am having a very difficult time understanding what this might look like. Can anyone give me a simple example that would display the principle pretty well? I'm having difficulty understanding the point or when/how this would be used or constructed. Any other forms of explanation are also welcome. Thank you.
 A: First, consider a simple, special case: $S = R$, and $\varphi = id$. Then for $a\in S$, $\varphi_a$  is just evaluation of a polynomial in $S[x]$ at $a$. Denote this by $e_a\colon S[x]\to S$, so
$$
e_a(\sum_i r_i x^i) = \sum_i r_i a^i
$$
$\varphi$ induces a homomorphism $\widehat\varphi\colon R[x]\to S[x]$ which transforms coefficients by $\varphi$:
$$
\widehat\varphi(\sum_i r_i x^i) \mapsto \sum_i \varphi(r_i) x^i
$$
Now, given $a\in S$, it's clear that $\varphi_a$ is just the composition of these two homomorphisms:
$$
\varphi_a\colon R[x]\stackrel{\widehat\varphi}\to S[x] \stackrel{e_a}\to S
$$
A: Consider $f:\mathbb{R}[x]\rightarrow \mathbb{R}$, given by $f(p(x))=p(0)$. So you are evaluating the value of the polynomial $p(x)$ at $x=0$. Similarly if you have a ring homomorphism $\phi:R\rightarrow S$ which sends $1_R$ to $1_S$, then take any $a\in S$ and any polynomial in co-efficients from $R$. Then using $\phi$ you can map the coefficients to elemnts in $S$ and for the indeterminate $x$, just "substitute" the value $a$ in it.
In this way, $\forall\ a\in S$, you get a $\rho$ map from $R[x]$ to $S$ (which is basically a valuation at $a$) such that $\rho$ coincides with $\phi$ on $R$.    
A: Given $f$ is a ring homomorphism from $R$  to $S$. We can fine an evaluation map $\psi_\alpha:R[x]\to R$,$\sum_i r_ix^i\mapsto \sum_ir_i\alpha^i$(which is also a ring homomorphism).
Now, we have
$R[x]\to^{\psi_{\alpha}} R\to^f S$.
Thus we have desire map $\Phi=fo\psi_\alpha:R[x]\to S$ that agrees on the coefficients of the polynomial and send $x\mapsto \alpha$.
