Proposition finlim_dfl_comp_cat_counit_laws : is_pstrans finlim_dfl_comp_cat_counit_data. Proof. refine (_ ,, _ ,, _). - refine (λ (C₁ C₂ : dfl_full_comp_cat) (F G : dfl_full_comp_cat_functor C₁ C₂) (τ : dfl_full_comp_cat_nat_trans F G), _). use dfl_full_comp_cat_nat_trans_eq. + abstract (intro x ; cbn ; rewrite id_left, id_right ; apply idpath). + intros x xx. use eq_cod_mor. rewrite transportf_cod_disp. cbn. exact (!(comp_cat_nat_trans_comprehension τ xx)). - refine (λ (C : dfl_full_comp_cat), _). use dfl_full_comp_cat_nat_trans_eq. + intro x ; cbn. apply idpath. + intros x xx. use eq_cod_mor. rewrite transportf_cod_disp. cbn. rewrite !id_left. refine (!_). exact (comprehension_functor_mor_id (comp_cat_comprehension C) xx). - refine (λ (C₁ C₂ C₃ : dfl_full_comp_cat) (F : dfl_full_comp_cat_functor C₁ C₂) (G : dfl_full_comp_cat_functor C₂ C₃), _). use dfl_full_comp_cat_nat_trans_eq. + intro x ; cbn. rewrite !id_left, !id_right. exact (!(functor_id _ _)). + intros x xx. use eq_cod_mor. rewrite transportf_cod_disp. cbn. rewrite !id_left, !id_right. refine (!_). etrans. { apply maponpaths. apply (comprehension_functor_mor_id (comp_cat_comprehension C₃)). } apply id_right. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseTerminal. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseEqualizers. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DependentSums. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.FullyFaithfulDispFunctor. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Examples.StructuredCategories. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat. Require Import UniMath.Bicategories.ComprehensionCat.DFLCompCat. Require Import UniMath.Bicategories.ComprehensionCat.ComprehensionEso. Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.FinLimToDFLCompCat. Require Import UniMath.Bicategories.ComprehensionCat.Biequivalence.DFLCompCatToFinLim.