Definition monbicat_inserter_ump_1_inv_2cell (q : inserter_cone Fm Gm) : invertible_2cell ((_,, fmonoidal_underlying_inserter_1cell q : monbicat ⟦q, monbicat_inserter_cone⟧) · inserter_cone_pr1 monbicat_inserter_cone) (inserter_cone_pr1 q). Proof. use tpair. - exists (pr112 (underlying_inserter_1cell q)). apply (is_mon_nat_trans_underlying_inserter_1cell_pr1 q). - use tpair. + use tpair. * exact (pr12 (inserter_1cell_pr1 (underlying_inserter_1cell q))). * apply is_mon_nat_trans_pointwise_inverse. apply (is_mon_nat_trans_underlying_inserter_1cell_pr1 q). + abstract (split ; use equality_2cells_monbicat ; intro ; apply id_left). Defined.
Require Import UniMath.Foundations.All. 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.Categories.Dialgebras. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.Bicategories.Limits.Inserters. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalDialgebras. Require Import UniMath.CategoryTheory.Monoidal.Displayed.MonoidalFunctorLifting. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.BicatOfWhiskeredMonCats. Require Import UniMath.Bicategories.Limits.Examples.BicatOfCatsLimits. Import BifunctorNotations. Import MonoidalNotations.