Definition identity_is_Pushout_input {a b c : C} {f : a --> b} {g : a --> c} (Pb : Pushout f g) : ∑ hk : Pb --> Pb, (PushoutIn1 Pb · hk = PushoutIn1 Pb) × (PushoutIn2 Pb · hk = PushoutIn2 Pb). Proof. exists (identity Pb). apply make_dirprod; apply id_right. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Core.Functors.