  We prove a correspondence between $\kappa$-small fibrations in simplicial
presheaf categories equipped with the injective or projective model structure
(and left Bousfield localizations thereof) and relatively $\kappa$-compact maps
in their underlying quasi-categories for suitably large regular cardinals
$\kappa$. We thus obtain a transition result between weakly universal small
fibrations in the (type theoretic) injective Dugger-Rezk-style standard
presentations of model toposes and object classifiers in Grothendieck
$\infty$-toposes in the sense of Lurie.
