What is a "set-like class"? Just / Weese contains the following theorem (p 126):
Theorem 5: Let $\mathbf Z \neq \varnothing$ and let $\mathbf W \subseteq \mathbf Z \times \mathbf Z$ be a wellfounded set-like class. Let $\mathbf G$ be a functional class such that $dom(\mathbf G)$ consists of all pairs of the form $\langle f,z \rangle$, where $z \in \mathbf Z$ and $f$ is a function with domain $I_{\mathbf W}(z)$. Then there exists exactly one functional class $\mathbf F : ]\mathbf Z \to \mathbf V$ such that $$ \mathbf F (z) = \mathbf G ( \mathbf F \mid I_{\mathbf W} (z), z) \hspace{0.5 cm} \forall z \in \mathbf Z$$
On the same page before the theorem they write "...In Chapter 12 we shall see that the existence of a definable, set-like wellorder of the universe is relatively consistent with $ZFC$. ..."
I am not aware of any other places in the book where the word "set-like" is used, in particular not a definition of what it means. Could someone give me the definition of "set-like" class and explain to me what it means? Many thanks for your help.
 A: I believe in this instance it should be "set-like well-founded relational class" or something like that.  The point is that for each $x \in \mathbf{Z}$ the class $\{ z \in \mathbf{Z} : z \mathrel{\mathbf{W}} x \}$ is actually a set.
Indeed, on p.125 Just-Weese give the following defintion:

A relational class $\mathbf{W}$ is said to be set-like if
  $$ (\forall x) (\exists y) ( y = \{ z : \langle z,x \rangle \in \mathbf{W} \} ).$$

Notice that this is precisely what you need in order to assert the existence of such a function(al class) $\mathbf{G}$ in the hypothesis of the theorem.  If $I_{\mathbf{W}} ( z )$ were a proper class for some $z \in \mathbf{Z}$, then there could be no pairs of the form $\langle f , z \rangle$ where $\mathrm{dom} ( f ) = I_{\mathbf{W}} (z)$, since $f$ must be a set, and so its domain must be a set, which it cannot be.  (Recall that only sets have real existence in ZF(C)).  Also, the function(al class) $\mathbf{F}$ that the theorem asserts to exist similarly could not exist because it could not be defined at such a $z$.
For an example of a well-founded relational class which fails to be set-like, consider $\mathbf{W} \subseteq \mathbf{On} \times \mathbf{On}$ defined by
$$\langle \alpha , \beta \rangle \in \mathbf{W} \quad \Longleftrightarrow \quad \begin{cases}
0 <  \alpha < \beta, &\text{or}\\
\beta = 0 \wedge \alpha \neq 0.
\end{cases}$$  This is a "well-ordering" of $\mathbf{On}$ which makes $0$ the unique maximum element (but otherwise retains the usual ordering of $\mathbf{On}$).  Clearly $I_{\mathbf{W}} ( 0 ) = \mathbf{On} \setminus \{ 0 \}$ is not a set.
