Proposition enriched_rezk_completion_ump_full_isaprop : isaprop (∑ (f : G₁ x --> G₂ x), ∏ (w : C₁) (i : z_iso (F w) x), τ w · #G₂ i = #G₁ i · f). Proof. use invproofirrelevance. intros φ₁ φ₂. use subtypePath. { intro. repeat (use impred ; intro). apply homset_property. } assert (H := HF₁ x). revert H. use factor_through_squash. { apply homset_property. } intros w. induction w as [ w i ]. use (cancel_z_iso' (functor_on_z_iso G₁ i)). exact (!(pr2 φ₁ w i) @ pr2 φ₂ w i). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.EnrichedCats.FullyFaithful. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.FunctorCategory. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.Precomposition. Require Import UniMath.CategoryTheory.Monoidal.Categories.