Definition functor_laxmon_equiv_layer {C D : bicat_of_univ_cats} (F : bicat_of_univ_cats⟦C,D⟧) (tuuaC : disp_univmon C) (tuuaD : disp_univmon D) : functor_lax_monoidal F (cmon_structure_from_layer tuuaC) (cmon_structure_from_layer tuuaD) ≃ tuuaC -->[F] tuuaD. Proof. use weq_iso. - apply functor_laxmon_to_layer. - apply functor_laxmon_from_layer. - intro ; apply idpath. - intro. use total2_paths_f. + apply idpath. + 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.