Proposition isaprop_preserves_fiberwise_subobject_classifier {C₁ C₂ : category} {F : C₁ ⟶ C₂} {D₁ : disp_cat C₁} {D₂ : disp_cat C₂} {HD₁ : cleaving D₁} {HD₂ : cleaving D₂} (TD₁ : fiberwise_terminal HD₁) (TD₂ : fiberwise_terminal HD₂) (FF : disp_functor F D₁ D₂) (HFF : preserves_fiberwise_terminal FF) : isaprop (preserves_fiberwise_subobject_classifier TD₁ TD₂ FF HFF). Proof. use impred ; intro. apply isaprop_preserves_subobject_classifier. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseTerminal. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifierIso. Require Import UniMath.CategoryTheory.SubobjectClassifier.PreservesSubobjectClassifier.