I am reading the book Quantum groups by Kassel. In proposition I.3.2 at the very beginning the reader is asked to show that under the identifications made, the maps $\Delta,\varepsilon$ and $S$ correspond to the maps $+,0$ and $-$.
However, $+$ is a map from $A^2$ to $A$, and I'm not sure how we can use the identifications to see that $\Delta:k[x]\rightarrow k[x',x'']$ corresponds to $+$.
It seems to me, all you can get from the identifications is that the map $\Delta$ corresponds to the element $x'+x''\in k[x',x'']$. Similarly, $\varepsilon$ corresponds to the element $0\in k$ and $S$ corresponds to the element $-x\in k[x]$.
So what exactly is meant in this proposition? I'm fairly sure this is a stupid question, but it's one that should be well-understood before proceeding any further in this theory.
Thank you in advance.