Lemma universalObjectFunctor_comm (C:category) {X Y:RepresentedFunctor C} (p:X-->Y) : p ⟳ universalElement (pr2 X) = universalElement (pr2 Y) ⟲ universalObjectFunctor C ▭ p. Proof. change (universalObjectFunctor C ▭ p) with (pr2 Y \\ (p ⟳ pr2 X)). apply pathsinv0, universalMapProperty. Defined.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import Require Import Require Import UniMath.MoreFoundations.Tactics.