proposition: Lemma ϕ₂_determined : ∏ x (f : a ⇒ x) , ϕ₂ i f = ϕ₂ i' f.
proof: 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.
imports: UniMath.Folds.UnicodeNotations UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.UnivalenceAxiom UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.Folds.folds_precat UniMath.Folds.from_precats_to_folds_and_back