In a Grothendieck $\infty$-topos, it is known that, for arbitrarily large regular cardinals $\kappa$, there is a classifier for the class of relatively $\kappa$-compact morphisms. It is also easy to show that this is not the case in 1-toposes, because we might have isomorphic, but not equal, such morphisms classified by the same map. However, we should be able to recover at least part of the definition of an object classifier. Namely, I need to know that in a Grothendieck 1-topos, for arbitrarily large regular cardinals $\kappa$, there is a map $t: U' \to U$ such that for every relatively $\kappa$-compact morphism $f: X \to Y$ there exists a pullback square


$\require{AMScd}$
\begin{CD}
X @>>> U'\\
@VfVV @VVtV\\
Y @>>> U
\end{CD}


(not necessarily unique and such that the map $Y \to U$ doesn't necessarily only classify $f$). I feel that this should definitely be true, but I can't find it anywhere in the literature. It would be very nice to have a reference for it, or a confutation in the unfortunate case I'm wrong. Thanks!



Object classifiers in 1-toposes
If $\mathcal{X}$ is a topos, then there is an adjunction $\mathcal{X} \leftrightarrows \mathcal{P(C)}$ with a category of presheaves of sets, where the right adjoint is fully faithful and accessible and the left is left exact.
The idea is very similar to the one spelled out in [https://www2.mathematik.tu-darmstadt.de/~streicher/NOTES/lift.pdf](https://www2.mathematik.tu-darmstadt.de/%7Estreicher/NOTES/lift.pdf), except that where its authors first find a suitable class of relatively $\kappa$-compact morphisms in $\mathcal{P(C)}$ and then consider the class of all morphisms in $\mathcal{X}$ that become relatively $\kappa$-compact in $\mathcal{P(C)}$, my analysis allows to find some $\kappa$ such that the desired class is that of all relatively $\kappa$-compact morphisms in $\mathcal{X}$. A complete proof can be found in Appendix A of <http://www.tac.mta.ca/tac/volumes/37/5/37-05.pdf>


I'd like to point out that neither strategy is more general than the other. The reason of this is that a class of relatively $\kappa$-compact morphisms in $\mathcal{X}$ need not be sent precisely to a class of relatively $\lambda$-compct morphisms in $\mathcal{P(C)}$ and, conversely, the preimage of a class of relatively $\lambda$-compact morphisms in $\mathcal{P(C)}$ need not be precisely a class of relatively $\kappa$-compact morphisms in $\mathcal{X}$.


However, all relatively $\kappa$-compact morphisms found in the latter proof are indeed sent to relatively $\kappa$-compact morphisms in $\mathcal{P(C)}$.


