proposition: Proposition from_isEqualizer_cod_fib_unique : isaprop (∑ (ζ : w --> e), ζ · ρ = h).
proof: Proof. use invproofirrelevance. intros ζ₁ ζ₂. use subtypePath. { intro. apply homset_property. } assert (H₁ : pr1 ζ₁ · cod_mor ee = h · f₁). { etrans. { apply maponpaths. exact (!(mor_eq ρr)). } rewrite !assoc. etrans. { apply maponpaths_2. exact (pr2 ζ₁). } apply idpath. } assert (H₂ : pr1 ζ₂ · cod_mor ee = h · f₁). { etrans. { apply maponpaths. exact (!(mor_eq ρr)). } rewrite !assoc. etrans. { apply maponpaths_2. exact (pr2 ζ₂). } apply idpath. } pose (ζ₁' := @make_cod_fib_mor _ _ w' ee (pr1 ζ₁) H₁). pose (ζ₂' := @make_cod_fib_mor _ _ w' ee (pr1 ζ₂) H₂). refine (maponpaths pr1 (EqualizerInsEq E ζ₁' ζ₂' _)). use eq_mor_cod_fib. rewrite !comp_in_cod_fib. exact (pr2 ζ₁ @ !(pr2 ζ₂)). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.PullbackConstructions UniMath.CategoryTheory.Limits.Preservation UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.FiberwiseTerminal UniMath.CategoryTheory.DisplayedCats.FiberwiseProducts UniMath.CategoryTheory.DisplayedCats.FiberwiseEqualizers UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod UniMath.CategoryTheory.DisplayedCats.Codomain.CodLeftAdjoint