This question is a follow-up on this question on Exercise 3.10 from Jech's Set Theory.
Given the hint, I understand that we seek a surjection from the subsets of $\omega_\alpha \times \omega_\alpha$ that are well-orderings to $\omega_{\alpha + 1}$. Asaf's answers in the linked question appears to provide such a surjection, but unfortunately I am not quite able to follow it. In particular, I cannot make sense of
So for each $\eta<\omega_{\alpha+1}$, there is some $R\subseteq\omega_\alpha\times\omega_\alpha$ which is a well-ordering of its field (domain and range) with order type $\eta$.
How, given a map $f_\eta: \eta \to \omega_\alpha$, is $R$ derived? And according to which well-order (I assume the canonical one?) on $\text{Ord} \times \text{Ord}$ is its order type $\eta$?
I would appreciate an elaboration on the quoted answer, in particular concerning the questions raised above.
Edit: (If relevant) I am doing self-studies in Set Theory following Jech's book and I am currently stuck at this question.
Given an injection $f:\eta \rightarrow \omega_\alpha$, define $R$ to be $R = \{(\alpha , \beta) \in ran(f)×ran(f): f^{-1}(\alpha) \lt f^{-1}(\beta)\}$. Now you can easily check that $R$ is a well-order on its field.