Definition orthogonal_via_pb_to_orthogonal (HB_2_1 : is_univalent_2_1 B) (Hf : orthogonal_via_pb HB_2_1) : f ⊥ g. Proof. apply (@adj_equiv_to_equiv_cat (univ_hom HB_2_1 b₂ c₁) (@univalent_iso_comma (univ_hom HB_2_1 b₁ c₁) (univ_hom HB_2_1 b₂ c₂) (univ_hom HB_2_1 b₁ c₂) (post_comp b₁ g) (pre_comp c₂ f)) (orthogonal_functor f g)). pose (pb_ump_mor_left_adjoint_equivalence _ _ (@iso_comma_has_pb_ump (univ_hom HB_2_1 b₁ c₁) (univ_hom HB_2_1 b₂ c₂) (univ_hom HB_2_1 b₁ c₂) (post_comp b₁ g) (pre_comp c₂ f)) Hf) as p. use (left_adjoint_equivalence_invertible p). - exact (orthogonal_via_pb_to_orthogonal_nat_trans HB_2_1). - use is_nat_z_iso_to_is_invertible_2cell. intro. use is_z_iso_iso_comma. + use is_inv2cell_to_is_z_iso ; cbn. is_iso. + use is_inv2cell_to_is_z_iso ; cbn. is_iso. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.IsoCommaCategory. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Properties.ClosedUnderInvertibles. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.PullbackFunctions. Import PullbackFunctions.Notations. Require Import UniMath.Bicategories.Limits.PullbackEquivalences. Require Import UniMath.Bicategories.Limits.Examples.BicatOfUnivCatsLimits.