Definition nattrans_strongmon_equiv_layer {C D : bicat_of_univ_cats} {F G : bicat_of_univ_cats⟦C,D⟧} (α : F ==> G) {tuuaC : disp_univstrongfunctor C} {tuuaD : disp_univstrongfunctor D} (ptuuaF : tuuaC -->[F] tuuaD) (ptuuaG : tuuaC -->[G] tuuaD) : nattrans_tensor_unit (functor_laxmon_from_layer _ _ _ (pr1 ptuuaF)) (functor_laxmon_from_layer _ _ _ (pr1 ptuuaG)) α ≃ ptuuaF ==>[α] ptuuaG. Proof. use weqimplimpl. - intro ptc. simpl. repeat split ; apply ptc. - intro ptc. simpl. repeat split ; apply ptc. - apply isaprop_nattrans_tensor_unit. - apply isaproptotal2 ; try (intro ; apply isapropunit). do 4 intro. use total2_paths_f. -- use total2_paths_f. { use total2_paths_f. - apply isaprop_preservestensor_commutes. - apply isaprop_preservesunit_commutes. } simpl. repeat (apply isapropdirprod) ; apply isapropunit. -- apply isapropunit. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.AssociatorUnitorsLayer. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.FinalLayer. Import Notations.