proposition: Proposition full_comp_nat_trans_eq {C₁ C₂ : full_comp_cat} {F G : full_comp_cat_functor C₁ C₂} {τ₁ τ₂ : full_comp_cat_nat_trans F G} (p : ∏ (x : C₁), τ₁ x = τ₂ x) (p' := nat_trans_eq (homset_property _) _ _ _ _ p) (pp : ∏ (x : C₁) (xx : disp_cat_of_types C₁ x), transportf _ (nat_trans_eq_pointwise p' x) (comp_cat_type_nat_trans τ₁ x xx) = comp_cat_type_nat_trans τ₂ x xx) : τ₁ = τ₂.
proof: Proof. use subtypePath. { intro. use isapropdirprod ; apply isapropunit. } use subtypePath. { intro. apply isaprop_comprehension_nat_trans_eq. } use subtypePath. { intro. use isapropdirprod ; apply isapropunit. } use total2_paths_f. - use nat_trans_eq. { apply homset_property. } exact p. - etrans. { use transportf_dirprod. } use dirprodeq. + simpl. rewrite transportf_const. apply (proofirrelevance _ (isapropdirprod _ _ isapropunit isapropunit)). + simpl. use disp_nat_trans_eq. intros x xx. etrans. { use disp_nat_trans_transportf. } apply pp. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Equivalences UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Preservation UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Examples.StructuredCategories UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.CompCat UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispAdjunctions UniMath.Bicategories.DisplayedBicats.DispInvertibles UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.Examples.Prod UniMath.Bicategories.DisplayedBicats.Examples.FullSub UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOfDispCats