Definition bincoprod_ump_2cell_invertible {a : B} {φ ψ : p --> a} (α : bincoprod_cocone_inl p · φ ==> bincoprod_cocone_inl p · ψ) (β : bincoprod_cocone_inr p · φ ==> bincoprod_cocone_inr p · ψ) (Hα : is_invertible_2cell α) (Hβ : is_invertible_2cell β) : is_invertible_2cell (bincoprod_ump_2cell α β). Proof. use make_is_invertible_2cell. - exact (bincoprod_ump_2cell (Hα^-1) (Hβ^-1)). - use (bincoprod_ump_2cell_unique (id2 _) (id2 _)). + abstract (rewrite <- !lwhisker_vcomp ; rewrite !bincoprod_ump_2cell_inl ; rewrite vcomp_rinv ; apply idpath). + abstract (rewrite <- !lwhisker_vcomp ; rewrite !bincoprod_ump_2cell_inr ; rewrite vcomp_rinv ; apply idpath). + abstract (apply lwhisker_id2). + abstract (apply lwhisker_id2). - use (bincoprod_ump_2cell_unique (id2 _) (id2 _)). + abstract (rewrite <- !lwhisker_vcomp ; rewrite !bincoprod_ump_2cell_inl ; rewrite vcomp_linv ; apply idpath). + abstract (rewrite <- !lwhisker_vcomp ; rewrite !bincoprod_ump_2cell_inr ; rewrite vcomp_linv ; apply idpath). + abstract (apply lwhisker_id2). + abstract (apply lwhisker_id2). 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.PrecategoryBinProduct. 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.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Section UniversalFromUMP. End UniversalFromUMP.