Definition indexed_functor_laws {C : category} {Φ Ψ : indexed_cat C} (τ : indexed_functor_data Φ Ψ) : UU := (∏ (x : C) (xx : Φ x), indexed_cat_id Ψ x (τ x xx) · indexed_functor_natural τ (identity x) xx = # (τ x) (indexed_cat_id Φ x xx)) × (∏ (x y z : C) (f : x --> y) (g : y --> z) (xx : Φ x), indexed_cat_comp Ψ f g (τ x xx) · indexed_functor_natural τ (f · g) xx = # (Ψ $ g) (indexed_functor_natural τ f xx) · (indexed_functor_natural τ g) ((Φ $ f) xx) · # (τ z) (indexed_cat_comp Φ f g xx)).
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.