proposition: Lemma CoproductArrow_diagram (Hti : I → Signature_category C D D') (Ht : Signature C D D') (F : ∏ i : I, SignatureMor C D D' (Hti i) Ht) X Y : Signature_category_mor_diagram C D D' (Sum_of_Signatures I C D D' CD Hti) Ht (CoproductArrow I _ (CCD _) (λ i, pr1 (F i))) X Y.
proof: Proof. apply Signature_category_mor_diagram_pointwise; intro c. etrans; [apply precompWithCoproductArrow|]. apply pathsinv0, CoproductArrowUnique; intro i; rewrite assoc; simpl. etrans; [apply cancel_postcomposition, (CoproductInCommutes _ _ _ (CD (λ j, pr1 (pr1 (pr1 (Hti j)) X) _)))|]. apply pathsinv0; etrans; [apply (nat_trans_eq_pointwise (pr2 (F i) X Y) c)|]. now etrans; [apply cancel_postcomposition, horcomp_id_left|]. Qed.
imports: UniMath.Foundations.PartD UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.PointedFunctors UniMath.CategoryTheory.PointedFunctorsComposition UniMath.CategoryTheory.HorizontalComposition UniMath.CategoryTheory.FunctorCategory UniMath.SubstitutionSystems.Notation UniMath.SubstitutionSystems.Signatures UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.SIP UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Core.Univalence UniMath.SubstitutionSystems.BinProductOfSignatures UniMath.SubstitutionSystems.SumOfSignatures