Definition pullback_preserves_biinitial (B : bicat_with_pb) (HI : strict_biinitial_obj B) {b₁ b₂ : B} (f : b₁ --> b₂) : preserves_biinitial (pb_psfunctor B f). Proof. pose (H := map_to_strict_biinitial_is_biinitial (pr2 HI) (pb_pr2 f (is_biinitial_1cell_property (pr12 HI) b₂))). use (preserves_chosen_biinitial_to_preserve_biinitial (_ ,, _) (pb_psfunctor B f)). - apply biinitial_in_slice. exact (pr1 HI ,, pr12 HI). - use (equiv_from_biinitial (is_biinitial_slice (pb_obj f (is_biinitial_1cell_property (pr12 HI) b₂) ,, H) b₁)). + use make_1cell_slice. * apply id₁. * cbn. use is_biinitial_invertible_2cell_property. exact H. + use left_adjoint_equivalence_in_slice_bicat. cbn. apply internal_adjoint_equivalence_identity. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. 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.Morphisms.Adjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Slice. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.PullbackFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction. Require Import UniMath.Bicategories.PseudoFunctors.Preservation.Preservation. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.Colimits.Initial. Require Import UniMath.Bicategories.Colimits.Extensive. Require Import UniMath.Bicategories.Colimits.Examples.SliceBicategoryColimits. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.PullbackFunctions.