Definition cocartesian_initial_weq_cocartesian_initial_via_adj (HB : is_univalent_2_1 B) : cocartesian_initial b ≃ cocartesian_initial_via_adj b T. Proof. use weqimplimpl. - exact cocartesian_initial_to_cocartesian_initial_via_adj. - exact cocartesian_initial_via_adj_to_cocartesian_initial. - apply isaprop_cocartesian_initial. exact HB. - apply isaprop_cocartesian_initial_via_adj. exact HB. 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.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Limits.Final. Require Import UniMath.Bicategories.Limits.Products. Section FromAdj. End FromAdj. Section FromAdj. End FromAdj.