I am studying algebraic curves through the book "Intr. to Algebraic Curves - Griffiths". The following definition puzzled too:
Definition: Suppose $C$ is a Riemann surface. Then a holomorphic differential $\omega$ is by definition a family ${(U_i, z_i, \omega_i)}$ such that
1)${(U_i,z_i)}$ is a holomorphic covering of $C$ and $\omega=f_i(z_i)dz_i$ where $f_i$ is holomorphic function on $U_i$,
2)if $z_i=\phi_{ij}(z_j)$ is a coordinate transformation on $U_i\cap U_j$, then
$f_i(\phi_{ij}(z_j))\frac{d\phi_{ij}(z_j)}{dz_j}=f_j(z_j)$
Question: Since domain of $f_i$ and $z_i$ are $U_i$; so how can we talk about $f_i(z_i)$?
Can one help to change the definition slightly?