Lemma coproduct_of_functors_nat_z_iso {I : UU} {C D : category} (CP : Coproducts I D) {F G : I → C ⟶ D} (α : ∏ i : I, nat_z_iso (F i) (G i)) : nat_z_iso (coproduct_of_functors I C D CP F) (coproduct_of_functors I C D CP G). Proof. exists (coproduct_of_functors_nat_trans CP α). intro c. use tpair. - use CoproductOfArrows. exact (λ i, pr1 (pr2 (α i) c)). - apply coproduct_of_functors_is_inverse. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Limits. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All.