proposition: Lemma isaprop_coh_strictness_from_univalence_2_1 (B : bicat) (HB : is_univalent_2_1 B) : isaprop (coh_strictness_structure B).
proof: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.TwoCategories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Categories.HSET.All UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.TransportLaws