Proposition to_bincoproduct_slice_unique : isaprop (∑ fg, ι₁ · fg = ρ₁ × ι₂ · fg = ρ₂). Proof. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. apply isapropdirprod ; apply homset_property. } use eq_mor_cod_fib. use (BinCoproductArrowsEq _ _ _ coprod). - cbn. refine (_ @ maponpaths dom_mor (pr12 φ₁ @ !(pr12 φ₂)) @ _). + rewrite comp_in_cod_fib. apply idpath. + rewrite comp_in_cod_fib. apply idpath. - cbn. refine (_ @ maponpaths dom_mor (pr22 φ₁ @ !(pr22 φ₂)) @ _). + rewrite comp_in_cod_fib. apply idpath. + rewrite comp_in_cod_fib. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod.