Definition tensor_after_pair_with_object_left {C : category} (T : functor (C × C) C) (I : C) : nat_z_iso (functor_fix_fst_arg _ _ _ T I) (functor_composite (pair_with_object_left I) T). Proof. use make_nat_z_iso. - exists (λ _, identity _). abstract (intro ; intros ; exact (id_right _ @ ! id_left _)). - intro x. exists (identity _). abstract (split ; apply id_right). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat.