Definition prebicategory_to_hcomp_bicat (B : prebicategory) : hcomp_bicat. Proof. simple refine (_ ,, _). - exact (prebicategory_to_hcomp_bicat_data B). - repeat split ; cbn ; intros. + apply id_left. + apply id_right. + apply assoc. + apply assoc'. + apply homset_property. + apply (functor_id ((pr221 (pr1 B)) b₁ b₂ b₃)). + apply (@functor_comp _ _ ((pr221 (pr1 B)) b₁ b₂ b₃) (_ ,, _) (_ ,, _) (_ ,, _) (_ ,, _) (_ ,, _)). + apply (@nat_trans_ax _ _ _ _ ((pr121 B) a b c d) (_ ,, (_ ,, _)) (_ ,, (_ ,, _)) (_ ,, (_ ,, _))). + apply (nat_trans_ax ((pr122 (pr1 B)) a b)). + apply (nat_trans_ax ((pr222 (pr1 B)) a b)). + apply (z_iso_inv_after_z_iso (make_z_iso _ _ ((pr112 B) b₁ b₂ b₃ b₄ f g h))). + apply (z_iso_after_z_iso_inv (make_z_iso _ _ ((pr112 B) b₁ b₂ b₃ b₄ f g h))). + apply (z_iso_inv_after_z_iso (make_z_iso _ _ ((pr121 (pr2 B)) b₁ b₂ f))). + apply (z_iso_after_z_iso_inv (make_z_iso _ _ ((pr121 (pr2 B)) b₁ b₂ f))). + apply (z_iso_inv_after_z_iso (make_z_iso _ _ ((pr221 (pr2 B)) b₁ b₂ f))). + apply (z_iso_after_z_iso_inv (make_z_iso _ _ ((pr221 (pr2 B)) b₁ b₂ f))). + apply B. + apply B. 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.