Let’s focus on $p=q=1$. By Proposition 4.4,

$$\frac{1}{4\pi}\frac{\partial^{2}w^{1\bar{1}}}{\partial\mu\partial\mu}+\frac{1}{\pi}Aa^{i\bar{j}}\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}}=-\delta(f_{S})|z_{1}|^{2}.$$

But by Lemma 4.6 we have $\frac{\partial^{2}w^{i\bar{j}}}{\partial\eta_{1}\partial\bar{\eta}_{1}}=\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}},$ so $Aa^{i\bar{j}}\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}}=\frac{\partial^{2}v}{\partial\eta_{1}\partial\bar{\eta}_{1}},$ hence the claim. ∎