Lemma Pullback_iso_squares_existence_comm₁ : Pullback_iso_squares_existence_mor · π'_y = g_y. Proof. simpl. refine (_ @ ! (z_iso_inv_on_left _ _ _ _ _ _ (pr221 t))). unfold Pullback_iso_squares_existence_mor. rewrite ! assoc'. apply maponpaths. use z_iso_inv_on_left. rewrite assoc'. apply pathsinv0. use z_iso_inv_on_right. exact π_iy. Qed.
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.