Definition equiv_from_biinitial : is_biinitial Y. Proof. use make_is_biinitial. - exact (λ Z, r · is_biinitial_1cell_property HX Z). - exact (λ Z f g, linvunitor _ • (ε^-1 ▹ f) • rassociator _ _ _ • (r ◃ is_biinitial_2cell_property HX _ (l · f) (l · g)) • lassociator _ _ _ • (ε ▹ g) • lunitor _). - abstract (intros Z f g α β ; use (vcomp_lcancel (lunitor _)) ; [ is_iso | ] ; rewrite <- !vcomp_lunitor ; apply maponpaths_2 ; use (vcomp_lcancel (ε ▹ f)) ; [ is_iso ; apply property_from_invertible_2cell | ] ; rewrite !vcomp_whisker ; apply maponpaths_2 ; use (vcomp_lcancel (lassociator _ _ _)) ; [ is_iso | ] ; rewrite <- !lwhisker_lwhisker ; apply maponpaths_2 ; apply maponpaths ; apply (is_biinitial_eq_property HX)). 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.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Univalence.