proposition: Lemma cat_prebicat_laws : prebicat_laws cat_prebicat_data.
proof: Proof. repeat split; cbn. - intros C D F G η. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_left. - intros C D F G η. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_right. - intros C D F₁ F₂ F₃ F₄ α β γ. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply assoc. - intros C₁ C₂ C₃ F G. apply nat_trans_eq; try apply homset_property. intro; apply idpath. - intros C₁ C₂ C₃ F G. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply functor_id. - intros C₁ C₂ C₃ F G₁ G₂ G₃ α β. apply nat_trans_eq; try apply homset_property. intros; apply idpath. - intros C₁ C₂ C₃ F₁ F₂ F₃ G α β. apply nat_trans_eq; try apply homset_property. intros ; cbn. exact (!(functor_comp G _ _)). - intros C D F G α. apply nat_trans_eq; try apply homset_property. intros ; cbn. rewrite id_left, id_right. apply idpath. - intros C D F G α. apply nat_trans_eq; try apply homset_property. intros ; cbn. rewrite id_left, id_right. apply idpath. - intros C₁ C₂ C₃ C₄ F G H₁ H₂ α. apply nat_trans_eq; try apply homset_property. intros ; cbn. rewrite id_left, id_right. apply idpath. - intros C₁ C₂ C₃ C₄ F G₁ G₂ H α. apply nat_trans_eq; try apply homset_property. intros ; cbn. rewrite id_left, id_right. apply idpath. - intros C₁ C₂ C₃ C₄ F₁ F₂ G H α. apply nat_trans_eq; try apply homset_property. intros ; cbn. rewrite id_left, id_right. apply idpath. - intros C₁ C₂ C₃ F₁ F₂ G₁ H₂ α β. apply nat_trans_eq; try apply homset_property. intros ; cbn. exact ((nat_trans_ax β _ _ _)). - intros C D F. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_left. - intros C D F. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_left. - intros C D F. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_left. - intros C D F. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_left. - intros C₁ C₂ C₃ C₄ F₁ F₂ F₃. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_left. - intros C₁ C₂ C₃ C₄ F₁ F₂ F₃. apply nat_trans_eq; try apply homset_property. intros ; cbn. apply id_left. - intros C₁ C₂ C₃ F G. apply nat_trans_eq; try apply homset_property. intros ; cbn. exact (id_left _ @ functor_id G _). - intros C₁ C₂ C₃ C₄ C₅ F₁ F₂ F₃ F₄. apply nat_trans_eq; try apply homset_property. intros ; cbn. rewrite !id_left. exact (functor_id F₄ _). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.catiso UniMath.CategoryTheory.CategoryEquality UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.Core.AdjointUnique UniMath.Bicategories.Core.Univalence