Definition preserves_subobject_classifier {C₁ C₂ : category} (F : C₁ ⟶ C₂) (T₁ : Terminal C₁) (T₂ : Terminal C₂) (HF : preserves_terminal F) : UU := ∏ (Ω : C₁) (t : T₁ --> Ω), is_subobject_classifier T₁ Ω t → is_subobject_classifier T₂ (F Ω) (TerminalArrow (preserves_terminal_to_terminal F HF T₁) _ · #F t).
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifierIso.