-- Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Basic
Goal:
C : Type u
inst : Category.{v, u} C
ℬ : (X Y : C) → LimitCone.{0, 0, v, u} (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
apply :
  ∀ (j : Discrete WalkingPair),
    Eq (α := tensorObj ℬ (tensorObj ℬ X₁ X₂) X₃ ⟶ Functor.obj.{0, v, 0, u} (pair Y₁ (ℬ Y₂ Y₃).cone.pt) j)
      (CategoryStruct.comp (Y := Cone.pt (F := pair Y₁ (ℬ Y₂ Y₃).cone.pt) (LimitCone.cone (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt)))
          (IsLimit.lift (LimitCone.isLimit (ℬ (tensorObj ℬ Y₁ Y₂) Y₃))
              (BinaryFan.mk (P :=
                Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃)))
                (CategoryStruct.comp (Y := tensorObj ℬ X₁ X₂)
                    (NatTrans.app (Cone.π (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃))) { as := WalkingPair.left } :
                      Functor.obj.{0, v, 0, u}
                          (Functor.obj (Functor.const (Discrete WalkingPair))
                            (Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃))))
                          { as := WalkingPair.left : Discrete WalkingPair } ⟶
                        Functor.obj.{0, v, 0, u} (pair (tensorObj ℬ X₁ X₂) X₃)
                          { as := WalkingPair.left : Discrete WalkingPair })
                    ((ℬ Y₁ Y₂).isLimit.lift
                      (BinaryFan.mk (P := (ℬ X₁ X₂).cone.pt)
                        ((ℬ X₁ X₂).cone.π.app { as := WalkingPair.left } ≫ f₁ : (ℬ X₁ X₂).cone.pt ⟶ Y₁)
                        ((ℬ X₁ X₂).cone.π.app { as := WalkingPair.right } ≫ f₂ : (ℬ X₁ X₂).cone.pt ⟶ Y₂))) :
                  Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃)) ⟶
                    tensorObj ℬ Y₁ Y₂)
                (NatTrans.app (Cone.π (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃))) { as := WalkingPair.right } ≫ f₃ :
                  Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃)) ⟶ Y₃)) ≫
            (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom)
          (NatTrans.app (Cone.π (LimitCone.cone (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt))) j) :
        tensorObj ℬ (tensorObj ℬ X₁ X₂) X₃ ⟶ Functor.obj.{0, v, 0, u} (pair Y₁ (ℬ Y₂ Y₃).cone.pt) j)
      (CategoryStruct.comp (Y := Cone.pt (F := pair Y₁ (ℬ Y₂ Y₃).cone.pt) (LimitCone.cone (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt)))
          ((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
            IsLimit.lift (LimitCone.isLimit (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)))
              (BinaryFan.mk (P := Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt)))
                (NatTrans.app (Cone.π (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt))) { as := WalkingPair.left } ≫ f₁ :
                  Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt)) ⟶ Y₁)
                (CategoryStruct.comp (Y := (ℬ X₂ X₃).cone.pt)
                    (NatTrans.app (Cone.π (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt))) { as := WalkingPair.right } :
                      Functor.obj.{0, v, 0, u}
                          (Functor.obj (Functor.const (Discrete WalkingPair))
                            (Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt))))
                          { as := WalkingPair.right : Discrete WalkingPair } ⟶
                        Functor.obj.{0, v, 0, u} (pair X₁ (ℬ X₂ X₃).cone.pt)
                          { as := WalkingPair.right : Discrete WalkingPair })
                    ((ℬ Y₂ Y₃).isLimit.lift
                      (BinaryFan.mk (P := (ℬ X₂ X₃).cone.pt)
                        ((ℬ X₂ X₃).cone.π.app { as := WalkingPair.left } ≫ f₂ : (ℬ X₂ X₃).cone.pt ⟶ Y₂)
                        ((ℬ X₂ X₃).cone.π.app { as := WalkingPair.right } ≫ f₃ : (ℬ X₂ X₃).cone.pt ⟶ Y₃))) :
                  Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt)) ⟶
                    tensorObj ℬ Y₂ Y₃)))
          (NatTrans.app (Cone.π (LimitCone.cone (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt))) j) :
        tensorObj ℬ (tensorObj ℬ X₁ X₂) X₃ ⟶ Functor.obj.{0, v, 0, u} (pair Y₁ (ℬ Y₂ Y₃).cone.pt) j)
⊢ Eq (α :=
    tensorObj ℬ (tensorObj ℬ X₁ X₂) X₃ ⟶
      Cone.pt (F := pair Y₁ (ℬ Y₂ Y₃).cone.pt) (LimitCone.cone (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt)))
    (IsLimit.lift (LimitCone.isLimit (ℬ (tensorObj ℬ Y₁ Y₂) Y₃))
          (BinaryFan.mk (P := Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃)))
            (CategoryStruct.comp (Y := tensorObj ℬ X₁ X₂)
                (NatTrans.app (Cone.π (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃))) { as := WalkingPair.left } :
                  Functor.obj.{0, v, 0, u}
                      (Functor.obj (Functor.const (Discrete WalkingPair))
                        (Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃))))
                      { as := WalkingPair.left : Discrete WalkingPair } ⟶
                    Functor.obj.{0, v, 0, u} (pair (tensorObj ℬ X₁ X₂) X₃)
                      { as := WalkingPair.left : Discrete WalkingPair })
                ((ℬ Y₁ Y₂).isLimit.lift
                  (BinaryFan.mk (P := (ℬ X₁ X₂).cone.pt)
                    ((ℬ X₁ X₂).cone.π.app { as := WalkingPair.left } ≫ f₁ : (ℬ X₁ X₂).cone.pt ⟶ Y₁)
                    ((ℬ X₁ X₂).cone.π.app { as := WalkingPair.right } ≫ f₂ : (ℬ X₁ X₂).cone.pt ⟶ Y₂))) :
              Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃)) ⟶
                tensorObj ℬ Y₁ Y₂)
            (NatTrans.app (Cone.π (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃))) { as := WalkingPair.right } ≫ f₃ :
              Cone.pt (F := pair (tensorObj ℬ X₁ X₂) X₃) (LimitCone.cone (ℬ (tensorObj ℬ X₁ X₂) X₃)) ⟶ Y₃)) ≫
        (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom :
      tensorObj ℬ (tensorObj ℬ X₁ X₂) X₃ ⟶
        Cone.pt (F := pair Y₁ (ℬ Y₂ Y₃).cone.pt) (LimitCone.cone (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt)))
    ((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
        IsLimit.lift (LimitCone.isLimit (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)))
          (BinaryFan.mk (P := Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt)))
            (NatTrans.app (Cone.π (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt))) { as := WalkingPair.left } ≫ f₁ :
              Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt)) ⟶ Y₁)
            (CategoryStruct.comp (Y := (ℬ X₂ X₃).cone.pt)
                (NatTrans.app (Cone.π (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt))) { as := WalkingPair.right } :
                  Functor.obj.{0, v, 0, u}
                      (Functor.obj (Functor.const (Discrete WalkingPair))
                        (Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt))))
                      { as := WalkingPair.right : Discrete WalkingPair } ⟶
                    Functor.obj.{0, v, 0, u} (pair X₁ (ℬ X₂ X₃).cone.pt)
                      { as := WalkingPair.right : Discrete WalkingPair })
                ((ℬ Y₂ Y₃).isLimit.lift
                  (BinaryFan.mk (P := (ℬ X₂ X₃).cone.pt)
                    ((ℬ X₂ X₃).cone.π.app { as := WalkingPair.left } ≫ f₂ : (ℬ X₂ X₃).cone.pt ⟶ Y₂)
                    ((ℬ X₂ X₃).cone.π.app { as := WalkingPair.right } ≫ f₃ : (ℬ X₂ X₃).cone.pt ⟶ Y₃))) :
              Cone.pt (F := pair X₁ (ℬ X₂ X₃).cone.pt) (LimitCone.cone (ℬ X₁ (ℬ X₂ X₃).cone.pt)) ⟶ tensorObj ℬ Y₂ Y₃)) :
      Cone.pt (F := pair (ℬ X₁ X₂).cone.pt X₃) (LimitCone.cone (ℬ (ℬ X₁ X₂).cone.pt X₃)) ⟶
        tensorObj ℬ Y₁ (tensorObj ℬ Y₂ Y₃))

Tactic:
apply (ℬ _ _).isLimit.hom_ext

Premises used: CategoryTheory.Limits.IsLimit.hom_ext, CategoryTheory.Limits.LimitCone.isLimit