Lemma inverse_is_inverse_2 a b (g : F a --> F b) : #F (inverse g) = g. Proof. unfold inverse. repeat rewrite functor_comp. rewrite functor_on_inv_from_z_iso. simpl. rewrite triangle_id_inverse'. rewrite <- assoc. set (H' := nat_trans_ax (adjcounit (pr1 H))). simpl in H'; rewrite H'; clear H'. rewrite assoc. set (H' := pathsinv0 (triangle_id_left_ad (pr2 (pr1 H)) a)). match goal with [|- ?f · ?g = ?h] => assert (H'' : identity _ = f) end. - simpl in *; apply H'. - rewrite <- H''. rewrite id_left. 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.