Lemma isotoid_functorcat_pointwise (C : precategory_data) (D : category) (H : is_univalent D) (F G : ob [C, D, D]) (α: z_iso F G) (c: C) : toforallpaths (fun _ : ob C => ob D) (pr1 (pr1 F)) (pr1 (pr1 G)) (maponpaths pr1 (maponpaths pr1 (isotoid (functor_category C D) (is_univalent_functor_category C D H) α))) c = isotoid D H (functor_z_iso_pointwise_if_z_iso' C D F G α c). Proof. assert (aux := isotoid_functorcat_pointwise_aux C D H F G (isotoid (functor_category C D) (is_univalent_functor_category C D H) α)). rewrite (idtoiso_isotoid [C, D] _ ) in aux. apply aux. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core.