proposition: Proposition preserves_binproduct_fiber_disp_codomain_functor {C₁ C₂ : category} (F : C₁ ⟶ C₂) (HF : preserves_pullback F) (x : C₁) : preserves_binproduct (fiber_functor (disp_codomain_functor F) x).
proof: Proof. intros y₁ y₂ z π₁ π₂ H. use isPullback_to_isBinProduct_cod_fib. apply isBinProduct_to_isPullback_cod_fib in H. use (isPullback_mor_paths _ _ _ _ _ _ (HF _ _ _ _ _ _ _ _ _ _ H)). - apply idpath. - apply idpath. - rewrite disp_codomain_fiber_functor_mor. apply idpath. - rewrite disp_codomain_fiber_functor_mor. apply idpath. - rewrite <- !functor_comp. apply maponpaths. exact (fib_isPullback π₁ π₂). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.Preservation UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations