Lemma pr1_pr1_relmonadmor_eq_from_relmonadmor_z_iso_idtoiso_aux {C : precategory_data} (D : category) (J : functor C D) {R R': RelMonad J} (p: @paths (category_RelMonad D J) R R'): base_paths (pr1 (R_functor R)) (pr1 (R_functor R')) (base_paths (R_functor R) (R_functor R') (maponpaths R_functor p)) = base_paths (pr1 R) (pr1 R') (base_paths R R' p). Proof. unfold base_paths. etrans. { apply maponpaths. apply (maponpathscomp (@R_functor C D J) (@functor_data_from_functor C D)). } etrans. { apply (maponpathscomp ((functor_data_from_functor C D ∘ R_functor)%functions) (@functor_on_objects C D)). } etrans. 2: { apply pathsinv0. apply (maponpathscomp (@RelMonad_data_from_RelMonad C D J) (@RelMonad_ob C D J)). } apply idpath. 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.