Definition iso_comma_ump1_preserves_pullback {C₀ : category} (H₁ : C₀ ⟶ C₁) (HH₁ : preserves_pullback H₁) (H₂ : C₀ ⟶ C₂) (HH₂ : preserves_pullback H₂) (α : nat_z_iso (H₁ ∙ F) (H₂ ∙ G)) : preserves_pullback (iso_comma_ump1 F G H₁ H₂ α). Proof. intros w x y z f g π₁ π₂ p₁ p₂ H. use isPullback_iso_comma ; cbn. - abstract (rewrite <- !functor_comp ; apply maponpaths ; exact p₁). - exact (HH₁ _ _ _ _ _ _ _ _ _ _ H). - abstract (rewrite <- !functor_comp ; apply maponpaths ; exact p₁). - exact (HH₂ _ _ _ _ _ _ _ _ _ _ H). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.IsoCommaCategory. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Preservation.