proposition: Proposition dfl_ext_identity_sub_mor_eq {Γ Δ : C} (s : Δ --> Γ) {A : ty Γ} (t₁ t₂ : comp_cat_tm Γ A) : dfl_ext_identity_sub_mor_mor s t₁ t₂ · # (fiber_functor_from_cleaving (disp_cat_of_types C) (cleaving_of_types C) s) (dfl_full_comp_cat_tm_to_mor t₁) = dfl_ext_identity_sub_mor_mor s t₁ t₂ · # (fiber_functor_from_cleaving (disp_cat_of_types C) (cleaving_of_types C) s) (dfl_full_comp_cat_tm_to_mor t₂).
proof: Proof. unfold dfl_ext_identity_sub_mor_mor. rewrite !assoc'. pose (EqualizerEqAr (dfl_ext_identity (sub_comp_cat_tm t₁ s) (sub_comp_cat_tm t₂ s))) as p. refine (_ @ p @ _) ; apply maponpaths ; clear p. - use (invmaponpathsweq (dfl_full_comp_cat_tm_weq_mor _)). refine (_ @ !(homotweqinvweq (dfl_full_comp_cat_tm_weq_mor _) _)). use eq_comp_cat_tm. cbn. rewrite comprehension_functor_mor_transportf. rewrite comprehension_functor_mor_comp. use (MorphismsIntoPullbackEqual (isPullback_Pullback (cartesian_to_pullback _ _))). + rewrite !assoc'. rewrite comprehension_functor_mor_factorisation_pr1. rewrite comprehension_functor_mor_transportf. rewrite comprehension_functor_mor_comp. rewrite PullbackArrow_PullbackPr1. etrans. { do 3 apply maponpaths. exact (maponpaths pr1 (FF_disp_functor_ff_inv (full_comp_cat_comprehension_fully_faithful C) _)). } cbn. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite comprehension_functor_mor_comm. rewrite !assoc. refine (_ @ id_left _). apply maponpaths_2. rewrite !assoc'. rewrite comprehension_functor_mor_comm. rewrite id_right. apply z_iso_after_z_iso_inv. + rewrite !assoc'. etrans. { do 2 apply maponpaths. apply comprehension_functor_mor_factorisation_pr2. } rewrite id_right. rewrite comprehension_functor_mor_comm. rewrite id_right. rewrite PullbackArrow_PullbackPr2. apply z_iso_after_z_iso_inv. - use (invmaponpathsweq (dfl_full_comp_cat_tm_weq_mor _)). refine (homotweqinvweq (dfl_full_comp_cat_tm_weq_mor _) _ @ _). use eq_comp_cat_tm. cbn. rewrite comprehension_functor_mor_transportf. rewrite comprehension_functor_mor_comp. use (MorphismsIntoPullbackEqual (isPullback_Pullback (cartesian_to_pullback _ _))). + rewrite !assoc'. rewrite comprehension_functor_mor_factorisation_pr1. rewrite comprehension_functor_mor_transportf. rewrite comprehension_functor_mor_comp. rewrite PullbackArrow_PullbackPr1. refine (!_). etrans. { do 3 apply maponpaths. exact (maponpaths pr1 (FF_disp_functor_ff_inv (full_comp_cat_comprehension_fully_faithful C) _)). } cbn. rewrite !assoc. apply maponpaths_2. rewrite !assoc'. rewrite comprehension_functor_mor_comm. rewrite !assoc. refine (_ @ id_left _). apply maponpaths_2. rewrite !assoc'. rewrite comprehension_functor_mor_comm. rewrite id_right. apply z_iso_after_z_iso_inv. + rewrite !assoc'. refine (!_). etrans. { do 2 apply maponpaths. apply comprehension_functor_mor_factorisation_pr2. } rewrite id_right. rewrite comprehension_functor_mor_comm. rewrite id_right. rewrite PullbackArrow_PullbackPr2. apply z_iso_after_z_iso_inv. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Limits.Preservation UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations 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.CategoryTheory.Monics UniMath.Bicategories.Core.Bicat UniMath.Bicategories.ComprehensionCat.BicatOfCompCat UniMath.Bicategories.ComprehensionCat.CompCatNotations UniMath.Bicategories.ComprehensionCat.DFLCompCat