Construction of *ZFC 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 ?
 A: It may be a typo, but you write $(\mathcal{U}_\gamma, \varepsilon_\gamma)$ is a direct product where the paper talks about a direct limit.
It is clearly important that the proof comes in two stages and the set $I$ is only needed for the first one. The second stage does not invoke an arbitrary collection of classes $(\mathcal{U}_i)_{i \in I}$, but a parameterised family $(\mathcal{U}_\alpha)_{\alpha \in \text{On}}$ indexed over the class of ordinals; there is a single formula $\Phi(x, \gamma)$ which identifies membership of $\mathcal{U}_\gamma$ whenever $\gamma$ is an ordinal.
Indeed $\Phi$ can express "there is a set $Y$ with elements indexed by $\gamma+1$ such that $Y_0$ verifies $Y_0 \in \text{V}^I_D$, for every $\delta < \gamma$ we have $Y_{\delta + 1}\, \subset\, Y_\delta$, for every limit $\delta \leq \gamma$ $Y_\delta$ is contained in the union of the previous elements of $Y$, and $x$ is equal to $Y_\gamma$."
Then the union $\mathcal{U}_D = \cup(\mathcal{U}_\alpha)$ is also a class, simply by quantifying out the parameter $\gamma$; $x \in \mathcal{U}_D$ iff $\exists \gamma \in \text{On} \, x \in \mathcal{U}_\gamma$.
I've left out the relations $\varepsilon_\gamma$, mostly out of laziness but the construction of the verifying formula mentioning them as well would proceed in exactly the same way. Up to laziness this should answer your question 1.
The critical point is that the $\varepsilon_\gamma$ are a class linked by end-extensions, with the property that any pair of them are always consistent on their joint domain. So we can relativise any relation $a\,\varepsilon\,b$ to any $\mathcal{U}_\gamma$ large enough that both elements appear, and it does not matter which.
For your question 2, we can take the successor $\mathcal{U}_{\gamma+1}$ of any $\mathcal{U}_\gamma$ in which $x$ exists, and the desired set will be in this successor and therefore in the union class $\mathcal{U}_D$.
Your question 3 again asks how a union of proper classes can be a class, and I am not sure how this differs from your question 1. Again the point is that the union is over a parametrised family, and the formula defining the union class is simply found by quantifying over the parameter. 
Question 4 asks why this union class satisfies Extensionality (as expressed with $\varepsilon$). I am a bit hazy on this---not having a month to work through it---but the answer must lie in the exact definition of $\varepsilon_\gamma$ in conjuction with the end-extensions $\chi$. Note that $\varepsilon_\gamma$ is defined in terms of the genuine subset operation $\subset$, so presumably Extensionality is inherited from the true property of ZFC.
