Proposition slice_constprod_functor1_mor_eq {C : category} (PB : Pullbacks C) {x : C} (BP := codomain_fib_binproducts PB x) (yf : C/x) {zg₁ zg₂ : C/x} (hp : zg₁ --> zg₂) : dom_mor (#(constprod_functor1 BP yf) hp) = slice_constprod_functor1_mor PB yf hp. Proof. unfold slice_constprod_functor1_mor. use (MorphismsIntoPullbackEqual (isPullback_Pullback _)). - rewrite PullbackArrow_PullbackPr1. etrans. { cbn -[fiber_category]. apply (PullbackArrow_PullbackPr1 (make_Pullback _ (pr22 (PB _ _ _ _ _)))). } rewrite id_right. cbn. apply idpath. - rewrite PullbackArrow_PullbackPr2. etrans. { cbn -[fiber_category]. apply (PullbackArrow_PullbackPr2 (make_Pullback _ (pr22 (PB _ _ _ _ _)))). } rewrite comp_in_cod_fib. cbn. apply idpath. Qed.
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.Adjunctions.Examples. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodDomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodColimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLeftAdjoint. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.whiskering. 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.StrictInitial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.DisjointBinCoproducts. Require Import UniMath.CategoryTheory.Limits.Preservation.