Lemma two_cat_to_prebicat_laws (C : two_cat) : prebicat_laws (two_cat_to_prebicat_data C). Proof. repeat split. - intros. apply TwoCategories.id2_left. - intros. apply TwoCategories.id2_right. - intros. apply TwoCategories.vassocr. - intros. apply TwoCategories.lwhisker_id2. - intros. apply TwoCategories.id2_rwhisker. - intros. apply TwoCategories.lwhisker_vcomp. - intros. apply TwoCategories.rwhisker_vcomp. - intros. apply TwoCategories.vcomp_lunitor. - intros. apply TwoCategories.vcomp_runitor. - intros. apply TwoCategories.lwhisker_lwhisker. - intros. apply TwoCategories.rwhisker_lwhisker. - intros. apply TwoCategories.rwhisker_rwhisker. - intros. apply TwoCategories.vcomp_whisker. - intros ; cbn. etrans. { apply idto2mor_comp. } rewrite pathsinv0r. apply idpath. - intros ; cbn. etrans. { apply idto2mor_comp. } rewrite pathsinv0l. apply idpath. - intros ; cbn. etrans. { apply idto2mor_comp. } rewrite pathsinv0r. apply idpath. - intros ; cbn. etrans. { apply idto2mor_comp. } rewrite pathsinv0l. apply idpath. - intros ; cbn. etrans. { apply idto2mor_comp. } rewrite pathsinv0r. apply idpath. - intros ; cbn. etrans. { apply idto2mor_comp. } rewrite pathsinv0l. apply idpath. - intros ; cbn. etrans. { apply maponpaths. apply idto2mor_rwhisker. } etrans. { apply idto2mor_comp. } refine (!_). etrans. { apply idto2mor_lwhisker. } refine (!_). apply maponpaths. apply (homset_property C). - intros ; cbn. etrans. { apply maponpaths. apply idto2mor_rwhisker. } etrans. { do 2 apply maponpaths_2. apply idto2mor_lwhisker. } etrans. { apply maponpaths_2. apply idto2mor_comp. } etrans. { apply idto2mor_comp. } refine (!_). etrans. { apply idto2mor_comp. } refine (!_). apply maponpaths. apply (homset_property C). 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.