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Call an internal poset $(P,\leqslant)$ internally inductive if for any object $X$ and any $C\rightarrowtail X\times P$ which is a chain of $X^*(P)$ in $\mathscr S/X$, the object $U_C\rightarrowtail X\times P$ of upper bounds of $C$ has support no less than $C$. That is, the image of the composite $U_C\rightarrowtail X\times P\to X$ contains the image of $C\rightarrowtail X\times P\to X$, where $X\times P\to X$ is the projection.

Call an internal poset $(P,\leqslant)$ internally inductive if for any object $X$ and any $C\rightarrowtail X\times P$ which is a chain of $X^*(P)$ in $\mathscr S/X$, the object $U_C\rightarrowtail X^*(P)$ of upper bounds of $C$ has support no less than $C$. That is, the image of the composite $U_C\rightarrowtail X\times P\to X$ contains the image of $C\rightarrowtail X\times P\to X$, where $X\times P\to X$ is the projection.

Here is, specifically, a version about which I would like to ask. This actually corresponds on the "classical" side to the variant of the Zorn's lemma with requiring upper bounds for nonempty chains only.

We then say that IZ' holds in $\mathscr S$ if for every internally inductive poset $P$, the object $\max(P)$ has the same support as $P$, as above.

Which toposes satisfy IZ'?

Here is a version about which I would like to ask specifically. This actually corresponds on the "classical" side to the variant of the Zorn's lemma with requiring upper bounds for nonempty chains only.

We then say that IIZ holds in $\mathscr S$ if for every internally inductive poset $P$, the object $\max(P)$ has the same support as $P$, as above.

Which toposes satisfy IIZ? How does it compare to IZ?

Important correction

Here is, specifically, a version about which I would like to ask. This actually corresponds on the "classical" side to the variant of the Zorn's lemma with requiring upper bounds for nonempty chains only.

Call an internal poset $(P,\leqslant)$ internally inductive if for any object $X$ and any $C\rightarrowtail X\times P$ which is a chain of $X^*(P)$ in $\mathscr S/X$, the object $U_C\rightarrowtail X\times P$ of upper bounds of $C$ has support no less than $C$. That is, the image of the composite $U_C\rightarrowtail X\times P\to X$ contains the image of $C\rightarrowtail X\times P\to X$, where $X\times P\to X$ is the projection.

We then say that IZ' holds in $\mathscr S$ if for every internally inductive poset $P$, the object $\max(P)$ has the same support as $P$, as above.

And the specific instance of my Extended question is,

Which toposes satisfy IZ'?

Important correction