Definition cartesian_1cell_pb_ump_1 : pb_ump_1 pb. Proof. intro q. pose (pr1 Hp x (make_ar (pb_cone_pr1 q)) (id₁ _) (pb_cone_pr2 q ,, comp_of_invertible_2cell (lwhisker_of_invertible_2cell _ (lunitor_invertible_2cell _)) (pb_cone_cell q))) as w. pose (lift_1cell_to_pb_1cell π p _ w) as r. use make_pb_1cell. - exact r. - exact (comp_of_invertible_2cell (pb_1cell_pr1 r) (runitor_invertible_2cell _)). - exact (pb_1cell_pr2 r). - abstract (refine (pb_1cell_eq r @ _) ; cbn ; rewrite <- !rwhisker_vcomp ; rewrite !vassocl ; apply maponpaths ; rewrite vassocl ; apply maponpaths ; rewrite !vassocr ; do 3 apply maponpaths_2 ; apply lunitor_lwhisker). 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.Core.Univalence. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. 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.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Core.Examples.OneTypes. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Codomain. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.Examples.OneTypesLimits.