Definition indexed_cat_data (C : category) : UU := ∑ (F₀ : C → univalent_category) (F₁ : ∏ (x y : C), x --> y → F₀ x ⟶ F₀ y), (∏ (x : C), functor_identity _ ⟹ F₁ x x (identity x)) × (∏ (x y z : C) (f : x --> y) (g : y --> z), F₁ x y f ∙ F₁ y z g ⟹ F₁ x z (f · g)).
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.