Lemma nat_trans_inv_pointwise_inv_after_p (C : precategory_data) (C' : precategory) (hs: has_homsets C') (F G : ob [C, C', hs]) (A : F --> G) (Aiso: is_iso A) a : inv_from_iso (functor_iso_pointwise_if_iso C C' hs F G A Aiso a) = pr1 (inv_from_iso (make_iso A Aiso)) a. Proof. apply pathsinv0. apply inv_iso_unique'. unfold precomp_with. simpl. set (TA := iso_inv_after_iso (make_iso A Aiso)). simpl in TA. apply (nat_trans_eq_pointwise TA). 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.