Lemma z_iso_to_isBinCoproduct {x y z : C} (BP : BinCoproduct x y) (i : z_iso z (BinCoproductObject BP)) : isBinCoproduct C _ _ z ((BinCoproductIn1 BP) · (z_iso_inv_from_z_iso i)) ((BinCoproductIn2 BP) · (z_iso_inv_from_z_iso i)). Proof. intros w f g. use unique_exists. - exact (i · (BinCoproductArrow BP f g)). - exact (z_iso_to_isBinCoproduct_comm BP i w f g). - abstract (intros y0; apply isapropdirprod; apply C). - abstract (intros y0 T; exact (z_iso_to_isBinCoproduct_unique BP i w f g y0 T)). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.ProductCategory.