Lemma is_mon_nat_trans_underlying_inserter_1cell_pr1 (q : inserter_cone Fm Gm) : is_mon_nat_trans (comp_fmonoidal (fmonoidal_underlying_inserter_1cell q) (dialgebra_monoidal_pr1 (pr2 Fm) (pr12 Gm))) (pr12 (inserter_cone_pr1 q)) (pr1 (inserter_1cell_pr1 (underlying_inserter_1cell q))). Proof. split. - intros x y. cbn. unfold projection_preserves_tensordata, fmonoidal_preservestensordata. rewrite id_left. rewrite id_right. etrans. 2: { apply cancel_postcomposition. apply pathsinv0, tensor_id_id. } apply pathsinv0, id_left. - red. cbn. unfold projection_preserves_unit, fmonoidal_preservesunit. rewrite id_left. apply id_right. Qed.
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.