proposition: Proposition preserves_pullback_from_binproduct_equalizer_unique : isaprop (∑ hk, (hk · # F (PullbackPr1 (PBC z x y f g)) = h) × (hk · # F (PullbackPr2 (PBC z x y f g)) = k)).
proof: Proof. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. apply isapropdirprod ; apply homset_property. } use (EqualizerInsEq FE). use (BinProductArrowsEq _ _ _ Fxy) ; cbn. - rewrite !assoc'. rewrite <- !functor_comp. exact (pr12 φ₁ @ !(pr12 φ₂)). - rewrite !assoc'. rewrite <- !functor_comp. exact (pr22 φ₁ @ !(pr22 φ₂)). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.PullbackConstructions UniMath.CategoryTheory.Limits.Preservation UniMath.CategoryTheory.Limits.EquivalencePreservation UniMath.CategoryTheory.Limits.LimitIso UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.CompositesAndInverses UniMath.CategoryTheory.whiskering