Lemma monads_category_Hstandard {F : functor C C} (T : disp_Monad_data F) (Tlaws : disp_Monad_laws T) (T' : disp_Monad_data F) (T'laws : disp_Monad_laws T') : disp_Monad_Mor_laws T T' (nat_trans_id F) → disp_Monad_Mor_laws T' T (nat_trans_id F) → T,, Tlaws = T',, T'laws. Proof. intros H H'. apply subtypeInjectivity. { intro T0. apply isaprop_disp_Monad_laws. } cbn. induction T as [μ η]. induction T' as [μ' η']. apply dirprodeq; cbn. - apply nat_trans_eq; [apply C|]. intro a. assert (Hinst := pr1 H a). cbn in Hinst. rewrite id_right in Hinst. rewrite id_left in Hinst. rewrite functor_id in Hinst. rewrite id_left in Hinst. exact Hinst. - apply nat_trans_eq; [apply C|]. intro a. assert (Hinst := pr2 H a). cbn in Hinst. rewrite id_right in Hinst. exact Hinst. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.SIP. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.Core.Univalence.