Definition preserves_chosen_coequalizer {C₁ C₂ : category} (HC₁ : Coequalizers C₁) (F : C₁ ⟶ C₂) : UU := ∏ (x y : C₁) (f g : x --> y) (p : # F f · # F (CoequalizerArrow (HC₁ x y f g)) = # F g · # F (CoequalizerArrow (HC₁ x y f g))), isCoequalizer (#F f) (#F g) (#F (CoequalizerArrow (HC₁ x y f g))) p.
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.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Adjunctions.Core.