proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.IndexedCategories.IndexedCategory UniMath.CategoryTheory.IndexedCategories.IndexedFunctor