-- Module: Mathlib.CategoryTheory.GradedObject.Trifunctor
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} C₃
inst_3 : Category.{v_4, u_4} C₄
inst_4 : Category.{v_5, u_5} C₁₂
inst_5 : Category.{v_6, u_6} C₂₃
F : C₁ ⥤ Functor.{v_6, v_4, u_6, u_4} C₂₃ C₄
G₂₃ : C₂ ⥤ Functor.{v_3, v_6, u_3, u_6} C₃ C₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₂₃ : BifunctorComp₂₃IndexData.{u_7, u_8, u_9, u_10, u_11} r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst_6 : (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p
inst_7 : (Functor.obj (Functor.obj (mapBifunctor F I₁ ρ₂₃.I₂₃) X₁) (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap ρ₂₃.q
H : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
j : J
c₂₃ : (i₂₃ : ρ₂₃.I₂₃) → (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).CofanMapObjFun ρ₂₃.p i₂₃ :=
  fun (i₂₃ : ρ₂₃.I₂₃) ↦ (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).cofanMapObj ρ₂₃.p i₂₃
h₂₃ : (i₂₃ : ρ₂₃.I₂₃) → IsColimit (c₂₃ i₂₃)
c : (Functor.obj (Functor.obj (mapBifunctor F I₁ ρ₂₃.I₂₃) X₁) (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).CofanMapObjFun ρ₂₃.q
  j :=
  (Functor.obj (Functor.obj (mapBifunctor F I₁ ρ₂₃.I₂₃) X₁) (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).cofanMapObj ρ₂₃.q j
hc : IsColimit c
c₂₃' : (i : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))) →
  Cocone (J := Discrete (↑(ρ₂₃.p ⁻¹' {(↑i : I₁ × ρ₂₃.I₂₃).2}) : Type (max u_8 u_9)))
    (Discrete.functor ((((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).mapObjFun ρ₂₃.p (↑i : I₁ × ρ₂₃.I₂₃).2) ⋙
      F.obj (X₁ (↑i : I₁ × ρ₂₃.I₂₃).1)) :=
  fun (i : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))) ↦
    (F.obj (X₁ (↑i : I₁ × ρ₂₃.I₂₃).1)).mapCocone (c₂₃ (↑i : I₁ × ρ₂₃.I₂₃).2)
hc₂₃' : (i : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))) → IsColimit (c₂₃' i)
Z : GradedObject (I₁ × I₂ × I₃) C₄ := (((mapTrifunctor (bifunctorComp₂₃ F G₂₃) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃
p' : I₁ × I₂ × I₃ → Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃ :=
  fun (x : I₁ × I₂ × I₃) ↦
    match (motive := I₁ × I₂ × I₃ → Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) x with
    | (i₁, i₂, i₃) => (i₁, ρ₂₃.p (i₂, i₃))
e : (i₁ : I₁) →
  (i₂₃ : ρ₂₃.I₂₃) →
    (↑(p' ⁻¹' {(i₁, i₂₃)}) : Type (max (max u_7 u_8) u_9)) ≃ (↑(ρ₂₃.p ⁻¹' {i₂₃}) : Type (max u_8 u_9)) :=
  fun (i₁ : I₁) (i₂₃ : ρ₂₃.I₂₃) ↦
    {
      toFun := fun (x : (↑(p' ⁻¹' {(i₁, i₂₃)}) : Type (max (max u_7 u_8) u_9))) ↦
        match (motive :=
          (↑(p' ⁻¹' {(i₁, i₂₃)}) : Type (max (max u_7 u_8) u_9)) → (↑(ρ₂₃.p ⁻¹' {i₂₃}) : Type (max u_8 u_9))) x with
        | ⟨(i₁', i₂, i₃), hi⟩ => Subtype.mk (p := fun (x : I₂ × I₃) ↦ x ∈ ρ₂₃.p ⁻¹' {i₂₃}) (i₂, i₃) ⋯,
      invFun := fun (x : (↑(ρ₂₃.p ⁻¹' {i₂₃}) : Type (max u_8 u_9))) ↦
        match (motive :=
          (↑(ρ₂₃.p ⁻¹' {i₂₃}) : Type (max u_8 u_9)) → (↑(p' ⁻¹' {(i₁, i₂₃)}) : Type (max (max u_7 u_8) u_9))) x with
        | ⟨(i₂, i₃), hi⟩ => Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦ x ∈ p' ⁻¹' {(i₁, i₂₃)}) (i₁, i₂, i₃) ⋯,
      left_inv := ⋯, right_inv := ⋯ }
x : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))
i₁ : I₁
i₂₃ : ρ₂₃.I₂₃
hi : (i₁, i₂₃) ∈ ρ₂₃.q ⁻¹' {j}
refine :
  Discrete.functor
      (Z.mapObjFun p'
        ((↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi) : I₁ × ρ₂₃.I₂₃).1,
          (↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi) :
              I₁ × ρ₂₃.I₂₃).2)) ≅
    (Discrete.equivalence (e i₁ i₂₃)).functor ⋙
      Discrete.functor
          ((((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).mapObjFun ρ₂₃.p
            (↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi) :
                I₁ × ρ₂₃.I₂₃).2) ⋙
        F.obj
          (X₁
            (↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi) :
                I₁ × ρ₂₃.I₂₃).1)
⊢ Z.CofanMapObjFun p'
    ((↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi) : I₁ × ρ₂₃.I₂₃).1,
      (↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi) : I₁ × ρ₂₃.I₂₃).2)

Tactic:
refine (Cocone.precompose (Iso.hom ?_)).obj ((Cocone.whiskeringEquivalence
        (Discrete.equivalence (e i₁ i₂₃))).functor.obj (c₂₃' ⟨⟨i₁, i₂₃⟩, hi⟩))

Premises used: CategoryTheory.Limits.Cocone.precompose, Prod.mk, CategoryTheory.Equivalence.functor, CategoryTheory.Discrete.equivalence, Subtype.mk, CategoryTheory.Limits.Cocone.whiskeringEquivalence, CategoryTheory.Iso.hom, CategoryTheory.Functor.obj