Lemma transportb_isotoid (C : category) (H : is_univalent C) (a b b' : ob C) (p : z_iso b b') (f : a --> b') : transportb (λ b0 : C, a --> b0) (isotoid C H p) f = f · inv_from_z_iso p. Proof. apply pathsinv0. apply transportb_transpose_right. change (precategory_morphisms a) with (λ b0 : C, a --> b0). rewrite transportf_isotoid'. rewrite <- assoc. rewrite z_iso_after_z_iso_inv. apply id_right. Qed.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos.