Proposition is_monic_dependent_product (φ : C /m Γ₁) : isMonic (cod_mor (R (mono_cod_to_cod_fib Γ₁ φ))). Proof. pose (right_adjoint_preserves_monics R (is_right_adjoint_right_adjoint _) (from_monic_slice_to_terminal (mono_cod_to_cod_fib _ φ) (codomain_fib_terminal _) (MonicisMonic _ (monic_of_mono_in_cat φ)))) as H. pose (make_Terminal _ (right_adjoint_preserves_terminal _ (HC _ _ s) _ (pr2 (codomain_fib_terminal _)))) as terminal_Γ₂. refine (to_monic_slice_from_terminal _ terminal_Γ₂ _). refine (transportf (λ z, isMonic z) _ H). apply (TerminalArrowEq (T := terminal_Γ₂)). Qed.
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.