Proposition local_equivalence_mnd_data_mor_laws : is_mnd_cell local_equivalence_mnd_data_mor_data. Proof. unfold is_mnd_cell ; cbn ; unfold local_equivalence_mnd_data_mor_data. rewrite !vassocl. etrans. { do 3 apply maponpaths. rewrite lwhisker_hcomp. rewrite <- linvunitor_natural. apply idpath. } rewrite !vassocr. apply maponpaths_2. rewrite vcomp_whisker. rewrite !vassocl. apply maponpaths. rewrite <- local_equivalence_inv_twocell_comp. rewrite !vassocl. rewrite <- vcomp_lunitor. etrans. { apply maponpaths. rewrite !vassocr. etrans. { apply maponpaths. apply maponpaths_2. exact (mnd_cell_endo τ). } rewrite !vassocl. apply local_equivalence_inv_twocell_comp. } rewrite !vassocr. apply maponpaths_2. use vcomp_move_R_pM ; [ is_iso | ]. rewrite !vassocr. use vcomp_move_L_Mp ; [ is_iso | ]. cbn. rewrite local_equivalence_twocell_rwhisker. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.DisplayedBicats.Examples.MonadsLax. Require Import UniMath.Bicategories.Monads.Examples.PsfunctorOnMonad. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Properties. Require Import UniMath.Bicategories.PseudoFunctors.LocalEquivalenceProperties. Require Import UniMath.Bicategories.PseudoFunctors.Examples.MonadInclusion.