Coordinate free description of a map induced by a base-point-free divisor on a compact Riemann surface

Question

Let $X$ be a compact Rieman surface, and $D$ be a divisor on $X$ with $|D|$ base-point-free. Note that the vector space $L(D)$ is finite-dimensional, and the complete linear system $|D|$ is naturally identified with the projectivization $\Bbb P(L(D))$. If we choose a basis $f_0,\dots,f_n$ for $L(D)$, then these meromorphic functions induce a map $\phi_D:X\to \Bbb P^n$ by $\phi_D(x)=[f_0(x):\cdots:f_n(x)]$ (and this map is unique up to coordinate change in $\Bbb P^n$). On the other hand, we have a map $\psi_D:X\to |D|^*=(\Bbb P(L(D)))^* \cong \Bbb P^n$ defined by sending $x\in X$ to the hyperplane $\{E\in |D|:E\geq x\} $ of the dual projective space $(\Bbb P(L(D)))^*$. In p.166 of Miranda's book Algebraic Curves and Riemann Surfaces, it is asserted that with suitable coordinates of $(\Bbb P(L(D)))^*$ (i.e. with a suitable identification of $(\Bbb P(L(D)))^*$ and $\Bbb P^n$), the map $\phi_D$ coincides with $\psi_D$, but I can't see why. What coordinates of $(\Bbb P(L(D)))^*$ should I choose?