proposition: Proposition hom_h_sq_precategory_laws : is_precategory hom_h_sq_precategory_data.
proof: Proof. use make_is_precategory_one_assoc. - intros h₁ h₂ s ; cbn. etrans. { do 2 apply maponpaths. apply lunitor_v_comp_square'. } rewrite <- rwhisker_square_comp. rewrite (linvunitor_lunitor (C := ver_bicat_of_verity_double_bicat)). rewrite rwhisker_square_id. rewrite <- lwhisker_square_comp. rewrite (linvunitor_lunitor (C := ver_bicat_of_verity_double_bicat)). rewrite lwhisker_square_id. apply idpath. - intros v₁ v₂ s ; cbn. etrans. { do 2 apply maponpaths. apply runitor_v_comp_square'. } rewrite <- rwhisker_square_comp. rewrite (lunitor_runitor_identity (C := ver_bicat_of_verity_double_bicat)). rewrite (rinvunitor_runitor (C := ver_bicat_of_verity_double_bicat)). rewrite rwhisker_square_id. rewrite <- lwhisker_square_comp. rewrite (runitor_lunitor_identity (C := ver_bicat_of_verity_double_bicat)). rewrite (linvunitor_lunitor (C := ver_bicat_of_verity_double_bicat)). rewrite lwhisker_square_id. apply idpath. - intros v₁ v₂ v₃ v₄ s₁ s₂ s₃ ; cbn. do 2 apply maponpaths. rewrite r_lwhisker_v_comp_square. rewrite r_rwhisker_v_comp_square. rewrite l_lwhisker_v_comp_square. rewrite l_rwhisker_v_comp_square. etrans. { do 2 apply maponpaths. apply lassociator_v_comp_square''. } rewrite !rwhisker_lwhisker_square. rewrite <- !rwhisker_square_comp. rewrite <- !lwhisker_square_comp. use eq_lwhisker_square. { rewrite lwhisker_hcomp. rewrite (triangle_l_inv (C := ver_bicat_of_verity_double_bicat)). rewrite <- rwhisker_hcomp. rewrite (lunitor_V_id_is_left_unit_V_id (C := ver_bicat_of_verity_double_bicat)). apply idpath. } use eq_rwhisker_square. { rewrite (lunitor_lwhisker (C := ver_bicat_of_verity_double_bicat)). apply maponpaths. apply (runitor_lunitor_identity (C := ver_bicat_of_verity_double_bicat)). } apply idpath. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Discrete UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Unitors UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.DoubleCategories.DoubleBicat.VerityDoubleBicat UniMath.Bicategories.DoubleCategories.DoubleBicat.DerivedLaws