Lemma alternative_inv_to_relmonadmor_z_iso_is_inv {C : precategory_data} (D : category) (J : functor C D) {R R': RelMonad J} (α : z_iso(C := category_RelMonad D J) R R'): alternative_inv_to_relmonadmor_z_iso D J α = inv_from_z_iso α. Proof. apply inv_z_iso_unique'. unfold precomp_with. apply RelMonadMor_equiv. - apply D. - apply funextsec. intro c. unfold alternative_inv_to_relmonadmor_z_iso. simpl. apply pathsinv0. apply z_iso_inv_on_left. rewrite id_left. intermediate_path (pr1 (pr1 α) c). { apply idpath. } rewrite z_iso_from_is_relmonadmor_z_iso_p. 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.