Definition identity_is_cartesian_2cell_sfib {x : B} {f g : x --> b} (α : f ==> g) : is_cartesian_2cell_sfib (id₁ b) α. Proof. intros h β δp r. use iscontraprop1. - abstract (use invproofirrelevance ; intros φ₁ φ₂ ; use subtypePath ; [ intro ; apply isapropdirprod ; apply cellset_property | ] ; use id1_faithful ; exact (pr12 φ₁ @ !(pr12 φ₂))). - refine (rinvunitor _ • δp • runitor _ ,, _ ,, _). + apply identity_help. + abstract (rewrite !vassocl ; rewrite <- vcomp_runitor ; use vcomp_move_R_pM ; [ is_iso | ] ; cbn ; rewrite <- vcomp_runitor ; rewrite !vassocr ; apply maponpaths_2 ; exact (!r)). 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.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.OpCellBicat. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration.