Definition algebra_as_coequalizer : Coequalizer ℓ ρ. Proof. use make_Coequalizer. - exact AX. - exact algebra_as_coequalizer_arr. - exact algebra_as_coequalizer_arr_eq. - intros Y g p. use iscontraprop1. + exact (algebra_as_coequalizer_ump_unique g). + simple refine (_ ,, _). * exact (algebra_as_coequalizer_ump g p). * exact (algebra_as_coequalizer_ump_comm g p). Defined.
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.Adjunctions.Core. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits. Require Import UniMath.CategoryTheory.Monads.Monads.