proposition: Proposition internal_pre_post_comp_comp {x₁ x₂ x₃ : V} (f₁ : x₁ --> x₂) (f₂ : x₂ --> x₃) {y₁ y₂ y₃ : V} (g₁ : y₁ --> y₂) (g₂ : y₂ --> y₃) : internal_pre_post_comp (f₁ · f₂) (g₁ · g₂) = internal_pre_post_comp f₂ g₁ · internal_pre_post_comp f₁ g₂.
proof: Proof. rewrite !internal_pre_post_comp_as_pre_post_comp. rewrite internal_precomp_comp. rewrite internal_postcomp_comp. rewrite !assoc'. apply maponpaths. rewrite !assoc. apply maponpaths_2. rewrite <- internal_pre_post_comp_as_pre_post_comp. rewrite <- internal_pre_post_comp_as_post_pre_comp. 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