Lemma comp_fmonoidal_unit_inv_laws : is_inverse_in_precat (fmonoidal_preservesunit (comp_fmonoidal_lax Fm Gm)) comp_fmnoidal_unit_inv. Proof. unfold comp_fmnoidal_unit_inv. split. - cbn. etrans. { rewrite !assoc'. apply maponpaths. rewrite assoc. apply cancel_postcomposition. rewrite <- functor_comp. apply maponpaths. apply (pr12 (fmonoidal_preservesunitstrongly Fm)). } rewrite functor_id. rewrite id_left. apply (pr12 (fmonoidal_preservesunitstrongly Gm)). - cbn. etrans. { rewrite !assoc'. apply maponpaths. rewrite assoc. apply cancel_postcomposition. apply (pr22 (fmonoidal_preservesunitstrongly Gm)). } rewrite id_left. rewrite <- functor_comp. etrans. { apply maponpaths. apply (pr22 (fmonoidal_preservesunitstrongly Fm)). } apply functor_id. 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.whiskering. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Import BifunctorNotations. Import MonoidalNotations.