Lemma is_exponentiable'_to_is_exponentiable'_app {y z : C} (f : y --> exp exp_x_alt_alt z) : exp_app_alt exp_x_alt_alt f = φ_adj_inv (pr2 exp_x_alt) f. Proof. refine (φ_adj_inv_under_iso _ _ _ (flip_z_iso _ _) _ _ _ _ @ _). refine (maponpaths (λ x, _ · x) (φ_adj_inv_under_iso _ _ _ _ _ _ _ _) @ _). refine (assoc _ _ _ @ _). refine (maponpaths (λ x, x · _) (z_iso_after_z_iso_inv (nat_z_iso_pointwise_z_iso (BinProduct_of_functors_commutes _ _ _ _ _) _)) @ _). apply id_left. Qed.
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.Adjunctions.Core. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.catiso. Require Import UniMath.CategoryTheory.CategoryEquality. Require Import UniMath.CategoryTheory.Core.Univalence.