Lemma inverse_is_inverse_1 a b (f : a --> b) : inverse (#F f) = f. Proof. unfold inverse. set (H' := nat_trans_ax (adjunit (pr1 H))). simpl in H'; rewrite <- H'; clear H'; simpl in *. rewrite <- assoc. intermediate_path (f · identity _). apply maponpaths. set (H' := z_iso_inv_after_z_iso (eta b)). apply H'. rewrite id_right. apply idpath. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core.