Definition make_isPullback {a b c d : C} (f : C ⟦b, a⟧) (g : C ⟦c, a⟧) (p1 : C⟦d,b⟧) (p2 : C⟦d,c⟧) (H : p1 · f = p2 · g) : (∏ e (h : C ⟦e, b⟧) (k : C⟦e,c⟧) (Hk : h · f = k · g ), ∃! hk : C⟦e,d⟧, (hk · p1 = h) × (hk · p2 = k)) → isPullback f g p1 p2 H. Proof. intros H' x cx k sqr. apply H'. assumption. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. 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.Core.Functors. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Monics. Definition UnivProductFromPullback (c d a : C) (f : a --> c) (g : a --> d): exists (UnivProductFromPullback c d a f g). Definition BinProductsFromPullbacks : BinProducts C := BinProduct_Pullback.