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

On notions of compactness, object classifiers and weak Tarski universes