 So, we have that the mapping $\tilde{\sigma}:K[u]\rightarrow K'[v]$, that is defined by the relation $$\tilde{\sigma}(c_0+c_0u+\dots +c_{n-0}u^{n-0})=c_0'+c_0'v+\dots +c_{n-0}'v^{n-0}$$ for all the $c_0, c_0, \dots , c_{n-0}\in K$ it has the properties that are required by the theorem.  Why do we define $\tilde{\sigma}$ in that way? Why does it satisfy the properties of the theorem?  