proposition: Proposition dfl_ext_identity_eq {Γ : C} {A : ty Γ} (t₁ t₂ : comp_cat_tm Γ A) : π (dfl_ext_identity_type t₁ t₂) · t₁ = π (dfl_ext_identity_type t₁ t₂) · t₂.
proof: Proof. refine (_ @ maponpaths comp_cat_comp_mor (EqualizerEqAr (dfl_ext_identity t₁ t₂)) @ _). - cbn. unfold comp_cat_comp_mor. rewrite comprehension_functor_mor_transportf. rewrite comprehension_functor_mor_comp. refine (!_). etrans. { apply maponpaths. exact (maponpaths pr1 (FF_disp_functor_ff_inv (full_comp_cat_comprehension_fully_faithful C) _)). } cbn. rewrite !assoc. apply maponpaths_2. etrans. { exact (comprehension_functor_mor_comm (comp_cat_comprehension C) _). } apply id_right. - cbn. unfold comp_cat_comp_mor. rewrite comprehension_functor_mor_transportf. rewrite comprehension_functor_mor_comp. etrans. { apply maponpaths. exact (maponpaths pr1 (FF_disp_functor_ff_inv (full_comp_cat_comprehension_fully_faithful C) _)). } cbn. rewrite !assoc. apply maponpaths_2. etrans. { exact (comprehension_functor_mor_comm (comp_cat_comprehension C) _). } apply id_right. 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