Lemma inv_of_f₂'_commutes_with_image_modulo_isos : # G f₂ · pr12 (z_iso_inv iy) = pr12 (z_iso_inv ix) · f₂'. Proof. unfold f₂. rewrite functor_on_fully_faithful_inv_hom. rewrite assoc'. etrans. { apply maponpaths. apply (z_iso_inv_after_z_iso (z_iso_inv iy)). } now rewrite id_right. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.WeakEquivalences.Core. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.WeakEquivalences.Reflection.Equalizers. Require Import UniMath.CategoryTheory.WeakEquivalences.Preservation.Equalizers.