Proposition indexed_nat_trans_natural_inv {C : category} {Φ Ψ : indexed_cat C} {τ θ : indexed_functor Φ Ψ} (m : indexed_nat_trans τ θ) {x y : C} (f : x --> y) (xx : Φ x) : m y ((Φ $ f) xx) · inv_from_z_iso (indexed_functor_natural_z_iso θ f xx) = inv_from_z_iso (indexed_functor_natural_z_iso τ f xx) · # (Ψ $ f) (m x xx). Proof. refine (!_). use z_iso_inv_on_right. rewrite !assoc. use z_iso_inv_on_left. exact (indexed_nat_trans_natural m f xx). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedCategory. Require Import UniMath.CategoryTheory.IndexedCategories.IndexedFunctor.