proposition: Lemma lineator_laxlaws_coprod : lineator_laxlaws Mon_V ActC ActD FF lineator_data_coprod.
proof: Proof. repeat split; red; intros; unfold lineator_data_coprod. - etrans. { repeat rewrite assoc. apply cancel_postcomposition. apply (lineator_linnatleft _ _ _ _ δll v). } repeat rewrite assoc'. apply maponpaths. etrans. { apply CoproductOfArrows_comp. } etrans. 2: { apply pathsinv0, CoproductOfArrows_comp. } apply maponpaths, funextsec; intro i; apply lineator_linnatleft. - etrans. { repeat rewrite assoc. apply cancel_postcomposition. apply (lineator_linnatright _ _ _ _ δll v1 v2 _ f). } repeat rewrite assoc'. apply maponpaths. etrans. { apply CoproductOfArrows_comp. } etrans. 2: { apply pathsinv0, CoproductOfArrows_comp. } apply maponpaths, funextsec; intro i; apply lineator_linnatright. - etrans. { rewrite assoc'. apply maponpaths. etrans. { apply CoproductOfArrows_comp. } cbn. apply maponpaths, funextsec; intro i; apply lineator_preservesactor. } etrans. { apply maponpaths. assert (aux : (fun i => aα^{ ActD }_{ v, w, F i x} · v ⊗^{ ActD}_{l} ll i w x · ll i v (w ⊗_{ ActC} x)) = (fun i => aα^{ ActD }_{ v, w, F i x} · (v ⊗^{ ActD}_{l} ll i w x · ll i v (w ⊗_{ ActC} x)))). { apply funextsec; intro i; apply assoc'. } rewrite aux. apply pathsinv0, CoproductOfArrows_comp. } etrans. { rewrite assoc. apply cancel_postcomposition. apply (lineator_preservesactor _ _ _ _ δll v w). } etrans. 2: { apply cancel_postcomposition. apply maponpaths. apply pathsinv0, (functor_comp (leftwhiskering_functor ActD v)). } repeat rewrite assoc'. do 2 apply maponpaths. repeat rewrite assoc. etrans. 2: { apply cancel_postcomposition. apply pathsinv0, (lineator_linnatleft _ _ _ _ δll v). } rewrite assoc'. apply maponpaths. etrans. 2: { apply pathsinv0, CoproductOfArrows_comp. } apply idpath. - etrans. 2: { apply (lineator_preservesunitor _ _ _ _ δll). } rewrite assoc'. apply maponpaths. etrans. { apply CoproductOfArrows_comp. } cbn. apply maponpaths, funextsec; intro i; apply lineator_preservesunitor. 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.PrecategoryBinProduct UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Actegories.Actegories UniMath.CategoryTheory.Actegories.MorphismsOfActegories UniMath.CategoryTheory.Actegories.ConstructionOfActegories UniMath.CategoryTheory.coslicecat UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Actegories.CoproductsInActegories UniMath.CategoryTheory.Actegories.ProductsInActegories UniMath.CategoryTheory.Actegories.ProductActegory