I'am reading 'A first course in modular form'. On page 221 Given two compact riemann surface and holomophic map $h$ : $X \to Y$.
Suppose $h$ is a surjection of finite degree $d$, that $h$ is locally $e_x$-to-1 at each $x \in X$ as already mentioned here, and that the set of exceptional points in $X$ is defined as $\mathcal{E}=\{x \in X$ : $\left.e_x>1\right\}$, the finite set of points where $h$ is ramified. Let $Y^{\prime}=Y-h(\mathcal{E})$ and $X^{\prime}=h^{-1}\left(Y^{\prime}\right)$ be the Riemann surfaces obtained by removing the images of the exceptional points from $Y$ and their preimages from $X$. The restriction of $h$ away from ramification and its image, $$ h: X^{\prime} \longrightarrow Y^{\prime}, $$ is a $d$-fold covering map where $d=\operatorname{deg}(h)$. This means that every point $y \in Y^{\prime}$ has a neighborhood $\widetilde{U}$ whose inverse image is a disjoint union of neighborhoods $U_1 \ldots . . U_d$ in $X^{\prime}$ such that each restriction $h_i: U_i \longrightarrow \widetilde{U}$ of $h$ is invertible.
The trace map induced by $h$ is a linear map transferring differentials from $X$ to $Y$, $$ \operatorname{tr}_h: \Omega_{\mathrm{hol}}^1(X) \longrightarrow \Omega_{\mathrm{hol}}^1(Y) . $$
If $\delta$ is a path in $Y^{\prime}$ lifting to a path in $X^{\prime}$ and $h_i^{-1}$ is a local inverse of $h$ about $\delta(0)$ taking $\delta(0)$ to the initial point of the lift then $h_i^{-1}$ has an analytic continuation along $\delta$, the chaining together of overlapping local inverses culminating in the local inverse about $\delta(1)$ taking $\delta(1)$ to the terminal point of the lift. To define the trace, let $\omega \in \Omega_{\mathrm{hol}}^1(X)$. Suppose $y$ is a point in $Y^{\prime}$, so that $h$ has local inverses $h_i^{-1}: \widetilde{U} \longrightarrow U_i, i=1, \ldots, d$. The trace is defined on $\widetilde{U}$ as the sum of local pullbacks, $$ \left.\left(\operatorname{tr}_h \omega\right)\right|_{\tilde{U}}=\sum_{i=1}^d\left(h_i^{-1}\right)^*\left(\left.\omega\right|_{U_i}\right) . $$
This local definition pieces together to a well defined global trace on $Y^{\prime}$ because analytically continuing the local inverses $h_i^{-1}$ along any loop in $Y^{\prime}$ back to $y$ permutes them, leaving the trace unaltered. The trace extends holomorphically from $Y^{\prime}$ to all of $Y$ (Exercise 6.2.4). The trace dualizes to a linear map of dual spaces, $$ \operatorname{tr}_h^{\wedge}: \Omega_{\text {hol }}^1(Y)^{\wedge} \longrightarrow \Omega_{\text {hol }}^1(X)^{\wedge}, \quad \operatorname{tr}_h^{\wedge} \psi=\psi \circ \operatorname{tr}_h . $$
The reverse change of variable formula for any path $\delta$ in $Y^{\prime}$, $$ \int_\delta\left(h^{-1}\right)^* \omega=\int_{h^{-1} \circ \delta} \omega, \quad \omega \in \Omega_{\mathrm{hol}}^1(X), $$ is meaningful so long as $h^{-1}$ is understood to be some local inverse of $h$ at $\delta(0)$ analytically continued along $\delta$, making $h^{-1} \circ \delta$ the lift of $\delta$ starting at $h^{-1}(\delta(0))$. This reduces to a finite sum of the same result on coordinate patches, where it is just forward change of variable (6.3) with $h^{-1}$ in place of $h$. Summing the reverse change of variable formula over local inverses gives for paths $\delta$ in $Y^{\prime}$ $$ \int_\delta \operatorname{tr}_h \omega=\sum_{\text {lifts } \gamma} \int_\gamma \omega, \quad \omega \in \Omega_{\text {hol }}^1(X) . $$
Definition 6.2.3. The reverse map of Jacobians is the holomorphic homomorphism induced by composition with the trace, $$ h^J: \operatorname{Jac}(Y) \longrightarrow \operatorname{Jac}(X), \quad h^J[\psi]=\left[\psi \circ \operatorname{tr}_h\right] . $$
Writing elements of $\operatorname{Jac}(Y)$ as sums of integrations per Abel's Theorem, the summed reverse change of variable formula (6.6) shows that the reverse maps of Jacobians transfers integration modulo homology from $Y$ to $X$ by pulling back the limits of integration with suitable multiplicity, $$ h^J\left(\sum_y n_y \int_{y_0}^y\right)=\sum_y n_y \sum_{x \in h^{-1}(y)} e_x \int_{x_0}^x . $$
My question is that why $h^J$ looks as this. From the last second formula, it should be integration along all the lifts of a given curve from $y_0$ to $y$, and I don't see why the right part is the integration along a fix point $x_0$ to other $x$. I think $x_0$ shoule change as the lifts.