proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Actegories.Actegories UniMath.CategoryTheory.Actegories.ConstructionOfActegories UniMath.CategoryTheory.coslicecat UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects