Definition has_binprod_cat_ump_binprod_ump_2_iscontr : iscontr (∑ (γ : φ ==> ψ), γ ▹ binprod_cone_pr1 p = α × γ ▹ binprod_cone_pr2 p = β). Proof. pose (pr1 (fully_faithful_implies_full_and_faithful _ _ _ (fully_faithful_from_equivalence _ _ _ (adj_equiv_to_equiv_cat _ (H x)))) φ ψ (α ,, β)) as m. use (factor_through_squash _ _ m). - apply isapropiscontr. - intro fib. use iscontraprop1. + apply has_binprod_cat_ump_binprod_ump_2_unique. + refine (pr1 fib ,, _ ,, _). * exact (maponpaths pr1 (pr2 fib)). * exact (maponpaths dirprod_pr2 (pr2 fib)). 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.