case w
C : Type u_2
inst✝⁵ : Category C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X : C
f : J → C
j✝ : J
⊢ biproduct.ι (fun j => X ⊗ f j) j✝ ≫ (leftDistributor X f).inv =
    biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)
C : Type u_2
inst✝⁵ : Category C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X : C
f : J → C
⊢ (leftDistributor X f).inv = ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)
ext
case w
C : Type u_2
inst✝⁵ : Category C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X : C
f : J → C
j✝ : J
⊢ (biproduct.ι (fun j => X ⊗ f j) j✝ ≫ biproduct.desc fun j => 𝟙 X ⊗ biproduct.ι f j) =
    biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)
case w
C : Type u_2
inst✝⁵ : Category C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X : C
f : J → C
j✝ : J
⊢ biproduct.ι (fun j => X ⊗ f j) j✝ ≫ (leftDistributor X f).inv =
    biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]
no goals
case w
C : Type u_2
inst✝⁵ : Category C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X : C
f : J → C
j✝ : J
⊢ (biproduct.ι (fun j => X ⊗ f j) j✝ ≫ biproduct.desc fun j => 𝟙 X ⊗ biproduct.ι f j) =
    biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)
simp only [Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp, zero_comp,
  Finset.sum_dite_eq, Finset.mem_univ, ite_true, eqToHom_refl, Category.id_comp,
  biproduct.ι_desc]