Proposition is_iso_span_associator_mor_eq : is_inverse_in_precat span_associator_mor span_associator_mor_inv. Proof. split. - use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. + rewrite id_left. unfold span_associator_mor_inv. rewrite !assoc'. rewrite PullbackArrow_PullbackPr1. unfold span_associator_mor. rewrite !assoc. rewrite !PullbackArrow_PullbackPr1. apply idpath. + rewrite id_left. unfold span_associator_mor_inv. rewrite !assoc'. rewrite PullbackArrow_PullbackPr2. use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. * rewrite !assoc'. rewrite PullbackArrow_PullbackPr1. unfold span_associator_mor. rewrite !assoc. rewrite PullbackArrow_PullbackPr1. rewrite PullbackArrow_PullbackPr2. apply idpath. * rewrite !assoc'. unfold span_associator_mor. rewrite !PullbackArrow_PullbackPr2. apply idpath. - use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. + rewrite id_left. unfold span_associator_mor. rewrite !assoc'. rewrite PullbackArrow_PullbackPr1. use (MorphismsIntoPullbackEqual (isPullback_Pullback (PC _ _ _ _ _))) ; cbn. * rewrite !assoc'. rewrite !PullbackArrow_PullbackPr1. unfold span_associator_mor_inv. rewrite PullbackArrow_PullbackPr1. apply idpath. * rewrite !assoc'. rewrite !PullbackArrow_PullbackPr2. unfold span_associator_mor_inv. rewrite !assoc. rewrite PullbackArrow_PullbackPr2. rewrite PullbackArrow_PullbackPr1. apply idpath. + rewrite !assoc'. unfold span_associator_mor. rewrite PullbackArrow_PullbackPr2. unfold span_associator_mor_inv. rewrite !assoc. rewrite !PullbackArrow_PullbackPr2. rewrite id_left. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Total. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Constant. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.DispCatOnTwoSidedDispCat.