Lemma full_pr1_category {C : category} (D : disp_cat C) (H : ∏ (a b : total_category D) (x : C ⟦ pr1 a, pr1 b ⟧), (∥ pr2 a -->[ x] pr2 b ∥)) : full (pr1_category D). Proof. intros ? ?. use pr1_issurjective. apply H. Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.AxiomOfChoice. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.