Algebraic group: Property of $f\in K[G]$.

Question

Let $G$ be an algebraic group with product $\mu:G\times G\to G$ and identity $e$.

We know that for $f\in K[G]$ then for $x,y\in G$ we have $\mu^*f=\sum f_i\otimes g_i$ for $f(xy)=\sum f_i(x)g_i(y)$.

Why do we have the property that: $$f=\sum g_i(e)f_i=\sum f_i(e)g_i?$$

I can't even interpret what this even means.

Extra tags: Humphreys - Linear Algebraic Groups - Page 71

Answer 1

Self answer, since I just got it - poor textbook notation in my opinion, but: $$f(x)=f(xe)=\sum f_i(x)g_i(e)$$ and $$f(x)=f(ex)=\sum f_i(e)g_i(x)$$

So $$f= \sum f_ig_i(e)=\sum f_i(e)g_i$$