Lemma dialgebra_nat_trans_is_mon_nat_trans : is_mon_nat_trans (comp_fmonoidal_lax dialgebra_monoidal_pr1 Fm) (comp_fmonoidal_lax dialgebra_monoidal_pr1 Gm) (dialgebra_nat_trans F G). Proof. split. + intros a a'. unfold fmonoidal_preservestensordata. cbn. unfold fmonoidal_preservestensordata. unfold projection_preserves_tensordata. cbn. unfold dialgebra_nat_trans_data. do 2 rewrite functor_id. do 2 rewrite id_right. unfold dialgebra_disp_tensor_op. etrans. { repeat rewrite assoc. do 2 apply cancel_postcomposition. apply (pr12 (fmonoidal_preservestensorstrongly Fm (pr1 a) (pr1 a'))). } rewrite id_left. apply idpath. + unfold is_mon_nat_trans_unitlaw. unfold fmonoidal_preservesunit. cbn. unfold fmonoidal_preservesunit. unfold projection_preserves_unit. cbn. unfold dialgebra_disp_unit. do 2 rewrite functor_id. do 2 rewrite id_right. etrans. { rewrite assoc. apply cancel_postcomposition. apply (pr12 (fmonoidal_preservesunitstrongly Fm)). } unfold fmonoidal_preservesunit. apply id_left. 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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Categories.Dialgebras. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.MonoidalSections. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Displayed.MonoidalFunctorLifting. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations. Import DisplayedBifunctorNotations. Import DisplayedMonoidalNotations. Section FromMonoidalSection. End FromMonoidalSection.