Lemma pointedtensorialstrength_preserves_actor {C : category} (M : monoidal C) {F : functor C C} (pts : pointedtensorialstrength M F) : preserves_actor _ _ _ _ pts ≃ ∏ (v w : coslice_cat_total C I_{M}) (x : C), pts (v ⊗_{monoidal_pointed_objects M} w) x · # F α^{M}_{pr1 v, pr1 w, x} = α^{M}_{pr1 v, pr1 w, F x} · pr1 v ⊗^{M}_{l} pts w x · pts v (pr1 w ⊗_{M} x). Proof. use weqimplimpl. - intros H v w x. assert (Hinst := H v w x). clear H. cbn in Hinst. unfold reindexed_actor_data in Hinst. cbn in Hinst. do 2 rewrite (bifunctor_rightid M) in Hinst. do 2 rewrite id_left in Hinst. exact Hinst. - intros H v w x. cbn. unfold reindexed_actor_data. cbn. do 2 rewrite (bifunctor_rightid M). do 2 rewrite id_left. apply H. - do 3 (apply impred; intro). apply C. - do 3 (apply impred; intro). apply C. 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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Import BifunctorNotations. Import MonoidalNotations. Import ActegoryNotations.