Definition postcomp_w_cart_refl_precart {C : category} {D : disp_cat C} {c c' c'' : C} {f : c' --> c} {f' : c'' --> c'} {d : D c} {d' : D c'} {d'' : D c''} (ff : d' -->[f] d) (ff' : d'' -->[f'] d') : is_cartesian ff -> (is_precartesian (ff' ;; ff) -> is_precartesian ff'). Proof. intros cartff precartff'ff. apply pre'_implies_pre. eapply postcomp_w_cart_refl_pre'cart. - exact cartff. - apply pre_implies_pre'. exact precartff'ff. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.MoreFibrations.Prefibrations.