Lemma cat_prebicat_nouniv_laws : prebicat_laws cat_prebicat_nouniv_data. Proof. split21; cbn. - intros C D F G η. apply nat_trans_eq; try apply D. intros ; cbn. apply id_left. - intros C D F G η. apply nat_trans_eq; try apply D. intros ; cbn. apply id_right. - intros C D F₁ F₂ F₃ F₄ α β γ. apply nat_trans_eq; try apply D. intros ; cbn. apply assoc. - intros C₁ C₂ C₃ F G. apply nat_trans_eq; try apply C₃. intro; apply idpath. - intros C₁ C₂ C₃ F G. apply nat_trans_eq; try apply C₃. intros ; cbn. apply functor_id. - intros C₁ C₂ C₃ F G₁ G₂ G₃ α β. apply nat_trans_eq; try apply C₃. intro; apply idpath. - intros C₁ C₂ C₃ F₁ F₂ F₃ G α β. apply nat_trans_eq; try apply C₃. intros ; cbn. exact (!(functor_comp G _ _)). - intros C D F G α. apply nat_trans_eq; try apply D. intros ; cbn. rewrite id_left, id_right. apply idpath. - intros C D F G α. apply nat_trans_eq; try apply D. 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 C₄. 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 C₄. 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 C₄. 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 C₃. intros ; cbn. exact ((nat_trans_ax β _ _ _)). - intros C D F. apply nat_trans_eq; try apply D. intros ; cbn. apply id_left. - intros C D F. apply nat_trans_eq; try apply D. intros ; cbn. apply id_left. - intros C D F. apply nat_trans_eq; try apply D. intros ; cbn. apply id_left. - intros C D F. apply nat_trans_eq; try apply D. intros ; cbn. apply id_left. - intros C₁ C₂ C₃ C₄ F₁ F₂ F₃. apply nat_trans_eq; try apply C₄. intros ; cbn. apply id_left. - intros C₁ C₂ C₃ C₄ F₁ F₂ F₃. apply nat_trans_eq; try apply C₄. intros ; cbn. apply id_left. - intros C₁ C₂ C₃ F G. apply nat_trans_eq; try apply C₃. intros ; cbn. exact (id_left _ @ functor_id G _). - intros C₁ C₂ C₃ C₄ C₅ F₁ F₂ F₃ F₄. apply nat_trans_eq; try apply C₅. intros ; cbn. rewrite !id_left. exact (functor_id F₄ _). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations.