Proposition to_monic_slice_from_terminal (H : isMonic (TerminalArrow T xf)) : isMonic (cod_mor xf). Proof. intros w g h p. simple refine (maponpaths pr1 (H (cod_fib_comp f (make_cod_fib_ob g)) (_ ,, _) (_ ,, _) _)). + cbn. rewrite id_right. apply idpath. + cbn. rewrite id_right. exact (!p). + apply TerminalArrowEq. 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.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.MonoCodomain. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.Sigma. Require Import UniMath.CategoryTheory.DisplayedCats.MonoCodomain.FiberMonoCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodLimits. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodDomain. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.