Definition sym_mon_cat_laws_tensored (V : monoidal_cat) (c : ∏ (x y : V), x ⊗ y --> y ⊗ x) : UU := (∏ (x y : V), c x y · c y x = identity _) × (∏ (x₁ x₂ y₁ y₂ : V) (f : x₁ --> x₂) (g : y₁ --> y₂), f #⊗ g · c x₂ y₂ = c x₁ y₁ · g #⊗ f) × (∏ (x y z : V), mon_lassociator x y z · c x (y ⊗ z) · mon_lassociator y z x = c x y #⊗ identity z · mon_lassociator y x z · identity y #⊗ c x z).
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.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Import MonoidalNotations.