proposition: Proposition is_iso_span_associator_mor_eq : is_inverse_in_precat span_associator_mor span_associator_mor_inv.
proof: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Total UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Constant UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.DispCatOnTwoSidedDispCat