Lemma monoidal_is_binproduct_from_comonoid (pI : εI = identity (monoidal_unit V)) (pT : ∏ x y : V, δ_{(x ⊗ y ,, m _) : comonoid V} · ((identity x #⊗ ε_{(y ,, m y) : comonoid V}) #⊗ (ε_{(x ,, m x) : comonoid V} #⊗ identity y) · mon_runitor x #⊗ mon_lunitor y) = identity (x ⊗ y)) : tensor_isBinProduct (monoidal_is_semicartesian_from_comonoid pI). Proof. intros x y. use make_isBinProduct. intros z fx fy. simple refine ((_ ,, _ ,, _) ,, _). - exact (make_isbinprod_from_comonoid_existence_mor fx fy). - exact (make_is_binprod_from_comonoids_existence_mor_1 pI fx fy). - exact (make_is_binprod_from_comonoids_existence_mor_2 pI fx fy). - intro f. use subtypePath. { intro ; apply isapropdirprod ; apply homset_property. } exact (make_is_binprod_from_comonoids_uniqueness fx fy (! pT x y) (pr1 f) (pr12 f) (pr22 f)). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import BifunctorNotations. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.SymmetricDiagonal. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Category. Require Import UniMath.CategoryTheory.Monoidal.Comonoids.Tensor. Import MonoidalNotations. Import ComonoidNotations. Import ComonoidNotations.