Definition nat_trans_to_cat_iso_inserter {C₁ C₂ C₃ : category} {F G : C₂ ⟶ C₃} {H₁ H₂ : C₁ ⟶ cat_iso_inserter F G} (α : H₁ ∙ cat_iso_inserter_pr1 F G ⟹ H₂ ∙ cat_iso_inserter_pr1 F G) (p : ∏ (x : C₁), pr12 (H₁ x) · # G (α x) = # F (α x) · pr12 (H₂ x)) : H₁ ⟹ H₂. Proof. use make_nat_trans. - exact (λ x, α x ,, p x). - abstract (intros x y f ; use eq_cat_iso_inserter ; cbn ; exact (nat_trans_ax α _ _ f)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.