Definition make_fiberwise_terminal_locally_propositional : fiberwise_terminal HD. Proof. split. - exact make_terminal_fiber_locally_propositional. - intros Γ₁ Γ₂ s. use preserves_terminal_if_preserves_chosen. + apply make_terminal_fiber_locally_propositional. + abstract (unfold preserves_chosen_terminal ; use (iso_to_Terminal (make_terminal_fiber_locally_propositional Γ₁)) ; use make_z_iso ; [ apply truth_sub | apply TerminalArrow | ] ; split ; apply HD'). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Section RightAdjointFromTerminal. End RightAdjointFromTerminal.