Lemma ϕ₂_determined : ∏ x (f : a ⇒ x) , ϕ₂ i f = ϕ₂ i' f. Proof. intros x f. rewrite (ϕ₂_is_comp i). rewrite (ϕ₂_is_comp i'). assert (H': is_inverse_in_precat (C:=C') (ϕ₁ i (id _ )) (ϕ₂ i' (id _ ))). { split. - rewrite H. apply ϕ₁_ϕ₂_id. - rewrite H. apply ϕ₂_ϕ₁_id. } assert (X : ϕ₂ i (id _ ) = ϕ₂ i' (id _ )). { set (H1:= inverse_unique_precat _ _ _ _ _ (ϕ₁_ϕ₂_are_inverse i) H'). assumption. } rewrite X. apply idpath. Qed.
Require Import UniMath.Folds.UnicodeNotations. Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Foundations.UnivalenceAxiom. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.Folds.folds_precat. Require Import UniMath.Folds.from_precats_to_folds_and_back.