Let $M,N$-smooth manifolds and $$\chi:M\rightarrow N$$ is a map between them. Then if we have a map $$f:N\rightarrow R$$Then the pullback of $f$ is the smooth function $\chi^*f$ defined by $$(\chi^*f)(x)=f(\chi(x))$$
If we have local coordinates $x^a$ on $M$ and $y^i$ on $N$ then the following is true $$\chi^*(y^i)=\chi^i(x)$$ And I have a question - where does this follow? I do not quite understand the meaning of the last line.