Proposition self_enrichment_equalizer_arr_eq {a b : V} (h : b --> a ⊸ v₁) (q : h · postcomp_arr (self_enrichment V) _ f = h · postcomp_arr (self_enrichment V) _ g) : h #⊗ identity a · internal_eval a v₁ · f = h #⊗ identity a · internal_eval a v₁ · g. Proof. rewrite !self_enrichment_postcomp in q. pose (maponpaths (λ z, z #⊗ identity _ · internal_eval _ _) q) as q'. cbn in q'. rewrite !tensor_comp_id_r in q'. rewrite !assoc' in q'. rewrite !internal_beta in q'. rewrite !assoc in q'. exact q'. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.SelfEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedTerminal. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedBinaryProducts. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedProducts. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedEqualizers. Require Import UniMath.CategoryTheory.EnrichedCats.Limits.EnrichedPowers. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Products. Import MonoidalNotations.