Definition mono_cod_implication_intro : binprod_in_fib (mono_codomain_fiberwise_binproducts PB) xφ mono_cod_implication --> yψ. Proof. use make_mono_cod_fib_mor. - refine (_ · mono_dom_mor (mono_cod_forall_intro φ mono_cod_implication_pb) · PullbackPr2 P). apply mono_cod_impl_pb_map. - abstract (cbn ; rewrite !assoc' ; etrans ; [ do 2 apply maponpaths ; refine (!(PullbackSqrCommutes P)) | ] ; etrans ; [ apply maponpaths ; rewrite !assoc ; apply maponpaths_2 ; apply (mono_mor_eq (mono_cod_forall_intro φ mono_cod_implication_pb)) | ] ; cbn ; rewrite !assoc ; unfold mono_cod_impl_pb_map ; rewrite PullbackArrow_PullbackPr2 ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Adjunctions.AdjunctionMonics. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.LocallyCartesianClosed.LocallyCartesianClosed. 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.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodMorphisms. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.MonoCodomain. Require Import UniMath.CategoryTheory.DisplayedCats.MonoCodomain.FiberMonoCod. Require Import UniMath.CategoryTheory.DisplayedCats.MonoCodomain.MonoCodLeftAdjoint. Require Import UniMath.CategoryTheory.DisplayedCats.MonoCodomain.MonoCodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.MonoCodomain.Inclusion. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DependentSums. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DependentProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseCartesianClosed. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DualBeckChevalley. Require Import UniMath.CategoryTheory.RegularAndExact.RegularCategory.