Definition path_cat_with_enrichment_alt_data_help_lemma {V : monoidal_cat} {ob : UU} {arr : ob -> ob -> V} {E₁ E₂ : cat_with_enrichment_alt_data_help ob arr} (p : pr1 E₁ = pr1 E₂) (q₁ : ∏ (x : ob), transportf (λ T, T x x) p (pr12 E₁ x) = pr12 E₂ x) (q₂ : ∏ (x y z : ob) (g₁ : pr1 E₁ x y) (g₂ : pr1 E₁ y z), transportf (λ T, T x z) p (pr122 E₁ _ _ _ g₁ g₂) = pr122 E₂ _ _ _ (transportf (λ T, T x y) p g₁) (transportf (λ T, T y z) p g₂)) (q₃ : ∏ (x y : ob) (g : pr1 E₁ x y), (pr1 (pr222 E₁)) _ _ g = (pr1 (pr222 E₂)) _ _ (transportf (λ T, T x y) p g)) (q₄ : ∏ (x y : ob) (g : I_{V} --> arr x y), transportf (λ T, T x y) p (pr2 (pr222 E₁) _ _ g) = pr2 (pr222 E₂) _ _ g) : E₁ = E₂.
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.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.EnrichedCats.Enriched.Enriched. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.Enriched.UnderlyingCategory. Import MonoidalNotations.