Lemma isaprop_coh_strictness_from_univalence_2_1 (B : bicat) (HB : is_univalent_2_1 B) : isaprop (coh_strictness_structure B). Proof. apply invproofirrelevance. intros S₁ S₂. use subtypePath. { exact (isPredicate_is_coh_strictness_structure HB). } use subtypePath. { exact (isPredicate_strictness_structure_laws B). } repeat use pathsdirprod. - use funextsec ; intro a. use funextsec ; intro b. use funextsec ; intro f. refine (_ @ isotoid_2_1_idtoiso_2_1 HB (plunitor S₂ f)). refine (!(isotoid_2_1_idtoiso_2_1 HB (plunitor S₁ f)) @ _). apply maponpaths. use subtypePath. { intro ; apply isaprop_is_invertible_2cell. } rewrite !idtoiso_plunitor. apply idpath. - use funextsec ; intro a. use funextsec ; intro b. use funextsec ; intro f. refine (_ @ isotoid_2_1_idtoiso_2_1 HB (prunitor S₂ f)). refine (!(isotoid_2_1_idtoiso_2_1 HB (prunitor S₁ f)) @ _). apply maponpaths. use subtypePath. { intro ; apply isaprop_is_invertible_2cell. } rewrite !idtoiso_prunitor. apply idpath. - use funextsec ; intro a. use funextsec ; intro b. use funextsec ; intro c. use funextsec ; intro d. use funextsec ; intro f. use funextsec ; intro g. use funextsec ; intro h. refine (_ @ isotoid_2_1_idtoiso_2_1 HB (plassociator S₂ f g h)). refine (!(isotoid_2_1_idtoiso_2_1 HB (plassociator S₁ f g h)) @ _). apply maponpaths. use subtypePath. { intro ; apply isaprop_is_invertible_2cell. } rewrite !idtoiso_plassociator. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.TwoCategories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.TransportLaws.