proposition: Proposition finlim_dfl_comp_cat_counit_laws : is_pstrans finlim_dfl_comp_cat_counit_data.
proof: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Equivalences.Core 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.Morphisms.Adjunctions UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.Examples.StructuredCategories UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Examples.Identity UniMath.Bicategories.PseudoFunctors.Examples.Composition UniMath.Bicategories.Transformations.PseudoTransformation UniMath.Bicategories.ComprehensionCat.BicatOfCompCat UniMath.Bicategories.ComprehensionCat.DFLCompCat UniMath.Bicategories.ComprehensionCat.ComprehensionEso UniMath.Bicategories.ComprehensionCat.Biequivalence.FinLimToDFLCompCat UniMath.Bicategories.ComprehensionCat.Biequivalence.DFLCompCatToFinLim