Lemma type_equiv_is_iso (A B : ob type_precat) (f : A ≃ B) : is_iso (C := type_precat) (pr1 f). Proof. apply (is_iso_qinv (C := type_precat) _ (invmap f)). split; simpl. - apply funextfun; intro; simpl in *. unfold compose, identity; simpl. apply homotinvweqweq. - apply funextfun; intro; simpl in *. unfold compose, identity; simpl. apply homotweqinvweq. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Categories.Type.Core.