Definition binprod_ump_2cell_invertible {a : B} {φ ψ : a --> p} {α : φ · binprod_cone_pr1 p ==> ψ · binprod_cone_pr1 p} {β : φ · binprod_cone_pr2 p ==> ψ · binprod_cone_pr2 p} (Hα : is_invertible_2cell α) (Hβ : is_invertible_2cell β) : is_invertible_2cell (binprod_ump_2cell α β). Proof. use make_is_invertible_2cell. - exact (binprod_ump_2cell (Hα^-1) (Hβ^-1)). - use (binprod_ump_2cell_unique (id2 _) (id2 _)). + abstract (rewrite <- !rwhisker_vcomp ; rewrite !binprod_ump_2cell_pr1 ; rewrite vcomp_rinv ; apply idpath). + abstract (rewrite <- !rwhisker_vcomp ; rewrite !binprod_ump_2cell_pr2 ; rewrite vcomp_rinv ; apply idpath). + abstract (apply id2_rwhisker). + abstract (apply id2_rwhisker). - use (binprod_ump_2cell_unique (id2 _) (id2 _)). + abstract (rewrite <- !rwhisker_vcomp ; rewrite !binprod_ump_2cell_pr1 ; rewrite vcomp_linv ; apply idpath). + abstract (rewrite <- !rwhisker_vcomp ; rewrite !binprod_ump_2cell_pr2 ; rewrite vcomp_linv ; apply idpath). + abstract (apply id2_rwhisker). + abstract (apply id2_rwhisker). 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.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.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Univalence.