Definition make_is_universal_inserter_cone_essentially_surjective : essentially_surjective (inserter_cone_functor cone x). Proof. intros h. use hinhpr. simple refine (_ ,, _). - exact (inserter_ump_mor H (pr1 h) (pr2 h)). - use z_iso_dialgebra. + simple refine (_ ,, _) ; cbn. * apply inserter_ump_mor_pr1. * abstract (cbn ; rewrite !vassocl ; use vcomp_move_R_pM ; [ is_iso | ] ; cbn ; rewrite !vassocr ; apply inserter_ump_mor_cell). + use is_inv2cell_to_is_z_iso. apply property_from_invertible_2cell. 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.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.Categories.Dialgebras. 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.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms.