Corollary equivalence_homtype_property (E : adj_equivalence_of_cats F) (P : UU → hProp) (prop : ∏ a b : ob C, P (C⟦a, b⟧)) : (∏ a b : ob D, P (D⟦a, b⟧)). Proof. apply ff_es_homtype_property. - apply fully_faithful_from_equivalence; assumption. - apply functor_from_equivalence_is_essentially_surjective; assumption. - assumption. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core.