In our background universe $V$ - satisfying $\operatorname{ZFC}$ - we say that an ordinal $\delta$ is a *Woodin cardinal* iff it satisfies one of the following equivalent properties:

1. For all $A \subseteq V_{\delta}$ there is a cardinal $\kappa < \delta$ such that for all $\nu < \delta$ there is a definable elementary embedding $j \colon V \prec M$, $M$ transitive, such that $\operatorname{crit}(j) = \kappa$, $j(\kappa) > \nu$ and $j(A) \cap V_{\nu} = A \cap V_{\nu}$. (Note that by reflecting $A^* := \{0\} \times A \cup \{1 \} \times V_{\delta}$ instead and choosing $\nu$ to be a cardinal, we can furthermore require that $V_{\nu} \subseteq M$.)
2. For all $A \subseteq V_{\delta}$ there is some $\kappa < \delta$ such that for all $\nu < \delta$ there is an elementary embedding as in 1. which is the canonical embedding $j = j_{E}$ of some extender $E$.
3. For every function $f \colon \delta \to \delta$ there is some $\kappa < \delta$ such that $f '' \kappa \subseteq \kappa$ and such that there is a definable elementary embedding $j \colon V \prec M$, $M$ transitive, with $\operatorname{crit}(j) = \kappa$ and $V_{j(f)(\kappa)} \subseteq M$.
4. For every function $f \colon \delta \to \delta$ there is some $\kappa < \delta$ with $f '' \kappa \subseteq \kappa$ and some extender $E$ with associated embedding $j_E \colon V \prec M$, $M$ transitive, such that $\operatorname{crit}(j_{E}) = \kappa$ and $V_{j_{E}(f)(\kappa)} \subseteq M$.

In [WHAT IS THE THEORY ZFC WITHOUT POWER SET?](https://arxiv.org/pdf/1110.2430v2.pdf) Victoria Gitman, Joel David Hamkins, and Thomas A. Johnstone showed - amonst other things - that in a model $N$ of $\operatorname{ZFC}_{-}$ (i.e. $\operatorname{ZFC}$ with the power set axiom removed) it is possible to have an embedding $j_{E} \colon N \to M$ associated to an $N$-extender $E \in N$ such that $M$ is transitive but $j_{E}$ is not elementary. I suspect that their construction can be generalized to produce a $\operatorname{ZFC}_{-}$-model $N$ in which 1. and 2. (resp. 3. and 4.) are not equivalent. 

[**Q:** Is there a model $N$ of $\operatorname{ZFC}_{-}$ in which 1. and 3. holds but 2. and 4. fail? (Or more generally (1. holds and 2. fails) or (3. holds and 4. fails).)]

I, however, am more interested in models of $\operatorname{ZFC}^{-}$  (i.e. $\operatorname{ZFC}$ with the power set axiom removed to which we add the axiom scheme of collection) [with a definable, global well-order]. Here (again demonstrated by Gitman, Hamkins and Thomas for $N$-measures - the same argument seems to work for $N$-extenders) Łoś's theorem does hold, so that a lack of elementarity doesn't stop us. Thinking through the proof it also seems that now 1. and 2. (resp. 3. and 4.) are equivalent. The remaining question is:

**Q:** Is there a model $N$ [with a definable, global well-order] of $\operatorname{ZFC}^{-}$ in which 1. and 3. are not equivalent for some $\delta \in N$? (I am also interested in the case that $N \models \mathcal{P}(\delta) \text{ does not exist}$.)