In the following paper, page 11 (Appendix), there is a construction of a model of a theory $^*ZFC$ (see the definitions in the paper included) from a model of $ZFC$. I have been trying really hard to understand it for almost a month without fruition. I have 4 questions.
The construction supposes that when we have a direct limit of $\in$-structures $(\mathcal{U}_i,\epsilon_i)_{i\in I}$ (where for all $i$ in $I$, $U_i$ is a class), with end-extensions $\chi_{i,j}$, then the direct limit $(\mathcal{U}_\gamma,\epsilon_\gamma)$
1/ is a class 2/ verifies the following property for $x$ in $U_\gamma$ : $\{x'\in U_\gamma | x'\epsilon_\gamma x\}$ is a set. How is it proved ?
Then the author procedes and defines the direct limit of classes indexed by the ordinals. 3/ The class of the ordinals being a proper class, how can one give a meaning to it ?
Finally, supposing that the structure $<U_d,\epsilon_d>$ exists, 4/ how can we prove that $<U_d,\epsilon_d>$ is a model of $ZFC^-$ ? First, the class does not seem to verify the extensionality axiom. The paper says the proof is "similar" to the proof of the fact that a transitive class $C$ that verifies "for all x which is a set, $x\subset C \Rightarrow x\in C$" is a model of $ZFC^-$. In what way is it similar ?