proposition: Proposition internal_precomp_comp {x₁ x₂ x₃ : V} (f₁ : x₁ --> x₂) (f₂ : x₂ --> x₃) (y : V) : internal_precomp (f₁ · f₂) y = internal_precomp f₂ y · internal_precomp f₁ y.
proof: Proof. use internal_funext. intros a h. rewrite tensor_split. rewrite tensor_comp_r_id_r. rewrite !assoc'. unfold internal_precomp. rewrite !internal_beta. refine (!_). rewrite tensor_split. rewrite !assoc'. apply maponpaths. rewrite !assoc. rewrite <- tensor_split'. rewrite tensor_split. rewrite !assoc'. rewrite internal_beta. rewrite !assoc. rewrite <- tensor_comp_id_l. apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Structure.Symmetric