Proposition is_strong_left_adjoint_on_unit_laws : is_inverse_in_precat (fmonoidal_preservesunit HL) is_strong_left_adjoint_on_unit. Proof. split ; unfold is_strong_left_adjoint_on_unit. - rewrite !assoc. exact (pr2 Hε). - rewrite !assoc'. etrans. { apply maponpaths. refine (!_). exact (nat_trans_ax ε _ _ (fmonoidal_preservesunit HL)). } cbn -[L R]. rewrite !assoc. rewrite <- functor_comp. refine (_ @ triangle_1_statement_from_adjunction A (monoidal_unit M₁)). apply maponpaths_2. apply maponpaths. refine (!(pr2 Hη) @ _). apply id_left. 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.Monads.Comonads. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Monoidal.FunctorCategories. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations. Section MonoidalAdjunctionFromStrong. End MonoidalAdjunctionFromStrong. Section SymMonoidalAdjunctionFromStrong. End SymMonoidalAdjunctionFromStrong.