evaluation map $$\Bev\!\equiv\!\prod_{j\in S}\ev_j: \M_{h,S}^*(X,\be;J)\lra X^S$$ is transverse to $\f$, (D\_[J;u\_]{}\^[NY]{})\_[jS]{}N\_[f\_j(w\_j)]{}\^[f\_j]{}Y, ((z\_j(u\_))+T\_[f\_j(w\_j)]{}Y+(d\_[w\_j]{}f\_j))\_[jS]{},is surjective; see Section \[AbsGW\_subs\] for the notation. Since $u_{\eff}$ is a regular map, $$\begin{split}
\dim \ker(D_{J;u_{\eff}}^{NY})
=\ind\big(D_{J;u_{\eff}}^{NY}\big)
&=2\big(\lr{c_1(NY),Y}+2(1\!-\!h)\big)=2\lr{c_1(TX),\be}\\
&=\sum_{j\in S}(4\!-\!\dim\,M_j)
\le\sum_{j\in S}\dim\,N^{f_j}_{f_j(w)}Y;\end{split}$$ the second-to-last equality holds by \_ref[dimcond\_e]{}. Thus, the homomorphism in \_ref[Ycount\_e]{} is an isomorphism. On the other hand, $D_{J;u}^{NY}$ is the restriction of the operator $\bigoplus_i D_{J;u_i}^{NY}$ to $$L^p_1(\Si_u;u^*NY)\subset\bigoplus_i L^p_1\big(\Si_{u;i};u_i^*NY\big),$$ where $\{\Si_{u;i}\}$ are the irreducible components of $\Si$ and $u_i\!=\!u|_{\Si_{u;i}}$. If $u_i$ is a constant map, then $D_{J;u_i}^{NY}$ is the usual $\dbar$-operator on the space of functions on $\Si_{u_i}$ with values in $N_{u_i(\Si_{u;i})}Y\!\approx\!\C^2$. Since $\Si_u$ is a connected nodal Riemann containing $\Si_{u_{\eff}}$ as a component, $u|_{\Si_{\eff}}\!=\!u_{\eff}$, and $u$ is constant on each of the irreducible components of $\Si_u\!-\!\Si_{u_{\eff}}$, it follows that the projection homomorphism D\_[J;u]{}\^[NY]{}D\_[J;u\_]{}\^[NY]{}, |\_[\_[u\_]{}]{},is an isomorphism. Thus, the homomorphism $$\ker(D_{J;u}^{NY})\lra \bigoplus_{j\in S}N_{f_j(w_j)}^{f_j}Y, \qquad
\xi\lra \big(\xi(z_j(u))+T_{f_j(w_j)}Y+(\Im\,d_{w_j}f_j)\big)_{j\in S}\,,$$ is an isomorphism, since the homomorphism \_ref[Ycount\_e]{} is. Therefore, by Proposition \[horreg\_prp\] there is no sequence in $$\ov\M_{g,\f}(X,\be;J)-\ov\M_{g,\f}(Y,[Y];J)\supset \M_{g,\f}^*(X,\be;J)$$ converging to $([u],(w_j)_{j\in S})$.\
We have thus shown that $\M_{g,\f}^*(X,\be;J)$ is a compact oriented $0$-dimensional manifold and its signed cardinality $E_{g,\be}^X(\f,J)$ is well-defined. The independence of $E_{g,\be}^X(\f,J)$ of the choices of $J$ and $f_j$ follows from \_ref[FanoGV\_e]{}, with $E_{h,\be}^X((\ka_j)_{j\in 