Definition preserves_underlying_laws {V₁ V₂ : monoidal_cat} {F : lax_monoidal_functor V₁ V₂} (Fv : preserve_underlying_data F) : UU := (∏ (v : V₁) (f : I_{V₂} --> F v), mon_functor_unit F · #F (Fv v f) = f) × (∏ (v : V₁) (f : I_{V₁} --> v), Fv v (mon_functor_unit F · #F f) = f).
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.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Import MonoidalNotations.