proposition: Proposition comprehension_eso_mor_inv_mor : comprehension_eso_mor · comprehension_eso_inv_mor = identity _.
proof: Proof. unfold comprehension_eso_mor, comprehension_eso_inv_mor. rewrite !assoc'. use z_iso_inv_on_right. rewrite id_right. refine (_ @ id_left _). rewrite !assoc. apply maponpaths_2. unfold comprehension_eso_mor_help, comprehension_eso_inv_mor_help. unfold sub_to_extension. use eq_sub_to_extension. - rewrite id_left. rewrite !assoc'. etrans. { do 4 apply maponpaths. exact (PullbackSqrCommutes (comp_cat_pullback comprehension_eso_ty_help comprehension_eso_inv_mor_help_sub)). } etrans. { do 3 apply maponpaths. rewrite !assoc. apply maponpaths_2. apply comp_cat_tm_eq. } rewrite id_left. unfold comprehension_eso_inv_mor_help_sub. refine (_ @ id_right _). use (MorphismsIntoPullbackEqual (isPullback_Pullback HΔ)). + rewrite !assoc'. rewrite (PullbackArrow_PullbackPr1 HΔ). rewrite z_iso_after_z_iso_inv. rewrite id_left, id_right. apply idpath. + rewrite !assoc'. rewrite (PullbackArrow_PullbackPr2 HΔ). rewrite id_left. use (cancel_z_iso _ _ γΓ). rewrite !assoc'. pose (maponpaths (λ z, z · PullbackPr1 HΓ) (!(dfl_ext_identity_eq left_tm right_tm))) as p. cbn -[left_tm right_tm PullbackPr1] in p. rewrite !assoc' in p. rewrite left_tm_pr1, right_tm_pr1 in p. rewrite !assoc' in p. cbn in p. exact p. - apply dfl_eq_subst_equalizer_tm. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.Preservation UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Equivalences UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.FiberwiseTerminal UniMath.CategoryTheory.DisplayedCats.FiberwiseProducts UniMath.CategoryTheory.DisplayedCats.FiberwiseEqualizers UniMath.CategoryTheory.DisplayedCats.DependentSums UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.FullyFaithfulDispFunctor UniMath.Bicategories.Core.Bicat UniMath.Bicategories.ComprehensionCat.BicatOfCompCat UniMath.Bicategories.ComprehensionCat.CompCatNotations UniMath.Bicategories.ComprehensionCat.DFLCompCat UniMath.Bicategories.ComprehensionCat.DFLCompCatNotations