Definition equalizer_of_setcategory_ump_mor_data {C₁ C₂ : setcategory} (F G : C₁ ⟶ C₂) {C₀ : setcategory} (H : C₀ ⟶ C₁) (p : H ∙ F = H ∙ G) : functor_data C₀ (equalizer_of_setcategory F G). Proof. use make_functor_data. - refine (λ x, H x ,, _). abstract exact (maponpaths (λ z, pr11 z x) p). - refine (λ x y f, #H f ,, _). abstract ( refine (_ @ path_functor_mor p f @ _); [ apply (maponpaths (λ x, _ · idtoiso x)); apply isaset_ob | apply (maponpaths (λ x, idtoiso x · _)); apply isaset_ob ] ). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.Limits.Equalizers.