Definition codomain_fiberwise_pullbacks (PB : Pullbacks C) (y : C) : Pullbacks (C/y). Proof. intros f₁ f₂ f₃ h₁ h₂. pose (pb := PB _ _ _ (dom_mor h₁) (dom_mor h₂)). use make_Pullback. - use make_cod_fib_ob. + exact pb. + exact (PullbackPr1 pb · cod_mor f₂). - use make_cod_fib_mor. + apply PullbackPr1. + abstract (apply idpath). - use make_cod_fib_mor. + apply PullbackPr2. + abstract (cbn ; rewrite <- (mor_eq h₁) ; rewrite <- (mor_eq h₂) ; rewrite !assoc ; apply maponpaths_2 ; exact (!(PullbackSqrCommutes pb))). - abstract (use eq_mor_cod_fib ; rewrite !comp_in_cod_fib ; cbn ; exact (PullbackSqrCommutes pb)). - use to_is_pullback_slice. + exact (PullbackSqrCommutes pb). + exact (isPullback_Pullback pb). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.PullbackConstructions. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseTerminal. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseEqualizers. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLeftAdjoint. Section FromEqCodFib. End FromEqCodFib. Section FromUMP. End FromUMP.