Lemma pr1_pr1_relmonadmor_eq_from_relmonadmor_z_iso_idtoiso {C : precategory_data} {D : category} (H: is_univalent D) (J : functor C D) {R R': RelMonad J} (p: @paths (category_RelMonad D J) R R'): pr1_pr1_relmonadmor_eq_from_relmonadmor_z_iso H J (idtoiso p) = base_paths (pr1 R) (pr1 R') (base_paths R R' p). Proof. unfold pr1_pr1_relmonadmor_eq_from_relmonadmor_z_iso. simpl. rewrite (z_iso_from_is_relmonadmor_z_iso_idtoiso D J). rewrite functor_eq_from_functor_z_iso_idtoiso. apply pr1_pr1_relmonadmor_eq_from_relmonadmor_z_iso_idtoiso_aux. 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.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core.