Definition is_strict_initial_parital_setoid : is_strict_initial initial_partial_setoid. Proof. intros X φ. use make_is_z_isomorphism. - exact (bot_partial_setoid_morphism X). - abstract (split ; [ apply bot_partial_setoid_morphism_eq_id | apply InitialArrowEq ]). Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.StrictInitial. Require Import UniMath.CategoryTheory.Hyperdoctrines.Hyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.FirstOrderHyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERs. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERMorphisms. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERCategory.