Lemma induced_precategory_reflects_pullbacks {M : category} {X:Type} (j : X -> ob M) {a b c d : induced_category M j} (f : b --> a) (g : c --> a) (p1 : d --> b) (p2 : d --> c) (H : p1 · f = p2 · g) : @isPullback _ _ _ _ _ (# (induced_precategory_incl j) f) (# (induced_precategory_incl j) g) (# (induced_precategory_incl j) p1) (# (induced_precategory_incl j) p2) H -> isPullback H. Proof. exact (λ pb T, pb (j T)). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Monics. Definition UnivProductFromPullback (c d a : C) (f : a --> c) (g : a --> d): exists (UnivProductFromPullback c d a f g). Definition BinProductsFromPullbacks : BinProducts C := BinProduct_Pullback.