Lemma nattrans_from_ms_is_mon_nat_trans : is_mon_nat_trans Fm Gm nattrans_from_ms. Proof. split. - intros a a'. assert (aux := smonoidal_preserves_tensor _ _ ms a a'). cbn in aux. unfold dialgebra_disp_tensor_op in aux. do 2 rewrite functor_id in aux. rewrite id_left, id_right in aux. unfold nattrans_from_ms. cbn. etrans. { apply maponpaths. apply pathsinv0, aux. } repeat rewrite assoc. apply cancel_postcomposition. etrans. { apply cancel_postcomposition. apply (z_iso_inv_after_z_iso (_,,fmonoidal_preservestensorstrongly Fm a a')). } apply id_left. - cbn. assert (aux := smonoidal_preserves_unit _ _ ms). cbn in aux. unfold dialgebra_disp_unit in aux. do 2 rewrite functor_id in aux. rewrite id_left, id_right in aux. etrans. { apply maponpaths. apply pathsinv0, aux. } rewrite assoc. etrans. { apply cancel_postcomposition. apply (z_iso_inv_after_z_iso (_,,fmonoidal_preservesunitstrongly Fm)). } 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.