Lemma is_inverse_nat_trans_inv_from_pointwise_inv (C : precategory_data)(C' : precategory) (hs: has_homsets C') (F G : [C, C', hs]) (A : F --> G) (H : is_nat_iso A) : is_inverse_in_precat A (nat_trans_inv_from_pointwise_inv C C' _ F G A H). Proof. simpl; split; simpl. - apply nat_trans_eq. apply hs. intro x; simpl. set (T := iso_inv_after_iso (tpair _ (pr1 A x) (H x))). apply T. - apply nat_trans_eq. apply hs. intro x; simpl. set (T := iso_after_iso_inv (tpair _ (pr1 A x) (H x))). apply T. Qed.
Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Univalence. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence.