Definition prebicategory_to_hcomp_bicat_data (B : prebicategory) : hcomp_bicat_data. Proof. simple refine (_ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _ ,, _). - exact (pr1 (pr111 B)). - intros b₁ b₂. exact (pr2 (pr111 B) b₁ b₂). - intros b₁ b₂ f g. exact (f --> g). - exact (pr1 (pr211 B)). - intros b₁ b₂ b₃ f g. exact (pr2 (pr211 B) b₁ b₂ b₃ (f ,, g)). - intros ; simpl. apply identity. - exact (λ _ _ _ _ _ α β, α · β). - cbn in *. intros b₁ b₂ f. exact (pr1 (pr2 (pr221 B) b₁ b₂) f). - intros a b f. apply (pr2 (pr212 B) _ _ f). - cbn in *. intros b₁ b₂ f. exact (pr1 (pr1 (pr221 B) b₁ b₂) f). - intros a b f. apply (pr1 (pr212 B) _ _ f). - intros b₁ b₂ b₃ b₄ f g h ; cbn in *. apply ((pr112 B) _ _ _ _ f g h). - intros b₁ b₂ b₃ b₄ f g h ; cbn in *. exact (pr1 ((pr121 B) _ _ _ _) (f ,, g ,, h)). - intros b₁ b₂ b₃ f₁ f₂ g₁ g₂ α β ; simpl in *. apply (#(pr2 (pr211 B) b₁ b₂ b₃)). exact (α ,, β). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.WkCatEnrichment.prebicategory. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws.