Definition montrafotargetbicat_associator: associator montrafotargetbicat_tensor. Proof. use make_nat_z_iso. + use make_nat_trans. * intro vηs. exists (monoidal_cat_associator Mon_V ((pr111 vηs,,pr121 vηs),,pr12 vηs)). apply montrafotargetbicat_associator_aux1. * intros vηs vηs' fgs. use total2_paths_f. -- cbn. repeat rewrite id_left. repeat rewrite id_right. exact (pr21 (monoidal_cat_associator Mon_V) ((pr111 vηs,, pr121 vηs),, pr12 vηs) ((pr111 vηs',, pr121 vηs'),, pr12 vηs') ((pr111 fgs,, pr121 fgs),, pr12 fgs)). -- apply trafotargetbicat_disp_cells_isaprop. + intro vηs. use make_is_z_isomorphism. * exists (pr1 (pr2 (monoidal_cat_associator Mon_V) ((pr111 vηs,, pr121 vηs),, pr12 vηs))). apply montrafotargetbicat_associator_aux2. * split. -- use total2_paths_f. ++ cbn. apply (pr2 (pr2 (monoidal_cat_associator Mon_V) ((pr111 vηs,, pr121 vηs),, pr12 vηs))). ++ apply trafotargetbicat_disp_cells_isaprop. -- use total2_paths_f. ++ cbn. apply (pr2 (pr2 (monoidal_cat_associator Mon_V) ((pr111 vηs,, pr121 vηs),, pr12 vηs))). ++ apply trafotargetbicat_disp_cells_isaprop. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.DisplayedMonoidalTensored. Require Import UniMath.Bicategories.MonoidalCategories.EndofunctorsMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.Actions. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrength. Require Import UniMath.Bicategories.MonoidalCategories.MonoidalFromBicategory. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Import Bicat.Notations.