Proposition faithful_weq_covyoneda_faithful (V : monoidal_cat) : faithful_moncat V ≃ faithful (covyoneda V (I_{V})). Proof. use weqimplimpl. - intros HV x y f. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. apply homset_property. } apply HV. intros g. exact (eqtohomot (pr2 φ₁ @ !(pr2 φ₂)) g). - intros HV x y f g H. use (invmaponpathsincl _ (HV x y) f g). use funextsec. exact H. - apply isaprop_faithful_moncat. - apply isaprop_faithful. 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.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.covyoneda. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Monoidal.Categories. Import MonoidalNotations.