-- 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 := ⋯ }
c₂₃'' : (i : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))) →
  Z.CofanMapObjFun p' ((↑i : I₁ × ρ₂₃.I₂₃).1, (↑i : I₁ × ρ₂₃.I₂₃).2) :=
  fun (x : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))) ↦
    match (motive :=
      (x : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))) →
        Z.CofanMapObjFun p' ((↑x : I₁ × ρ₂₃.I₂₃).1, (↑x : I₁ × ρ₂₃.I₂₃).2))
      x with
    | ⟨(i₁, i₂₃), hi⟩ =>
      Functor.obj (C :=
        Cocone
          ((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)))
        (Cocone.precompose
          (Iso.hom
            (Discrete.natIso
              fun
                (x :
                  Discrete
                    (↑(p' ⁻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).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)}) :
                      Type (max (max u_7 u_8) u_9))) ↦
              match (motive :=
                (x :
                    Discrete
                      (↑(p' ⁻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).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)}) :
                        Type (max (max u_7 u_8) u_9))) →
                  Functor.obj
                      (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)))
                      x ≅
                    ((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)).obj
                      x)
                x with
              | { as := ⟨(i₁', i₂, i₃), hi_1⟩ } => eqToIso ⋯)))
        (Functor.obj (Equivalence.functor (Cocone.whiskeringEquivalence (Discrete.equivalence (e i₁ i₂₃))))
          (c₂₃' (Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi)))
h₂₃'' : (i : (↑(ρ₂₃.q ⁻¹' {j}) : Type (max u_7 u_11))) → IsColimit (c₂₃'' i)
refine :
  ∀ (x : Discrete (↑(r ⁻¹' {j}) : Type (max (max u_7 u_8) u_9))) (i₁ : I₁) (i₂ : I₂) (i₃ : I₃)
    (h : (i₁, i₂, i₃) ∈ r ⁻¹' {j}),
    Eq (α :=
      Functor.obj (Discrete.functor (Z.mapObjFun r j))
          { as := Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦ x ∈ r ⁻¹' {j}) (i₁, i₂, i₃) h :
            Discrete (↑(r ⁻¹' {j}) : Type (max (max u_7 u_8) u_9)) } ⟶
        (cofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j).pt)
      (CategoryStruct.comp (Y :=
        Cocone.pt
          (Z.cofanMapObjComp p' ρ₂₃.q r ⋯ j
            (fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) (h : ρ₂₃.q x = j) ↦
              match (motive := (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) → (h : ρ₂₃.q x = j) → Z.CofanMapObjFun p' x) x, h with
              | (i₁, i₂₃), h =>
                c₂₃'' (Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) h))
            c))
        (NatTrans.app
            (Cocone.ι
              (Z.cofanMapObjComp p' ρ₂₃.q r ⋯ j
                (fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) (h : ρ₂₃.q x = j) ↦
                  match (motive := (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) → (h : ρ₂₃.q x = j) → Z.CofanMapObjFun p' x) x,
                    h with
                  | (i₁, i₂₃), h =>
                    c₂₃'' (Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) h))
                c))
            { as := Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦ x ∈ r ⁻¹' {j}) (i₁, i₂, i₃) h } :
          Functor.obj (Discrete.functor (Z.mapObjFun r j))
              { as := Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦ x ∈ r ⁻¹' {j}) (i₁, i₂, i₃) h :
                Discrete (↑(r ⁻¹' {j}) : Type (max (max u_7 u_8) u_9)) } ⟶
            Functor.obj
              (Functor.obj (Functor.const (Discrete (↑(r ⁻¹' {j}) : Type (max (max u_7 u_8) u_9))))
                (Cocone.pt
                  (Z.cofanMapObjComp p' ρ₂₃.q r ⋯ j
                    (fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) (h : ρ₂₃.q x = j) ↦
                      match (motive := (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) → (h : ρ₂₃.q x = j) → Z.CofanMapObjFun p' x) x,
                        h with
                      | (i₁, i₂₃), h =>
                        c₂₃'' (Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) h))
                    c)))
              { as := Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦ x ∈ r ⁻¹' {j}) (i₁, i₂, i₃) h :
                Discrete (↑(r ⁻¹' {j}) : Type (max (max u_7 u_8) u_9)) })
        (Iso.hom
          (Iso.refl
            (Cocone.pt
              (Z.cofanMapObjComp p' ρ₂₃.q r ⋯ j
                (fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) (h : ρ₂₃.q x = j) ↦
                  match (motive := (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) → (h : ρ₂₃.q x = j) → Z.CofanMapObjFun p' x) x,
                    h with
                  | (i₁, i₂₃), h =>
                    c₂₃'' (Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) h))
                c)))))
      ((cofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j).ι.app
        { as := Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦ x ∈ r ⁻¹' {j}) (i₁, i₂, i₃) h })
⊢ IsColimit (cofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j)

Tactic:
refine IsColimit.ofIsoColimit (isColimitCofanMapObjComp Z p' ρ₂₃.q r ρ₂₃.hpq j
    (fun ⟨i₁, i₂₃⟩ h => c₂₃'' ⟨⟨i₁, i₂₃⟩, h⟩) (fun ⟨i₁, i₂₃⟩ h => h₂₃'' ⟨⟨i₁, i₂₃⟩, h⟩) c hc)
    (Cocone.ext (Iso.refl _) (fun ⟨⟨i₁, i₂, i₃⟩, h⟩ => ?_))

Premises used: CategoryTheory.GradedObject.isColimitCofanMapObjComp, CategoryTheory.Limits.Cocone.ext, CategoryTheory.Limits.IsColimit.ofIsoColimit, CategoryTheory.Limits.IsColimit, CategoryTheory.Iso.refl, CategoryTheory.GradedObject.isColimitCofan₃MapBifunctor₁₂BifunctorMapObj.match_12, Prod.mk, CategoryTheory.GradedObject.isColimitCofan₃MapBifunctorBifunctor₂₃MapObj.match_9, CategoryTheory.GradedObject.BifunctorComp₂₃IndexData.hpq, CategoryTheory.GradedObject.BifunctorComp₂₃IndexData.q, Subtype.mk, CategoryTheory.Limits.Cocone.ι, CategoryTheory.GradedObject.CofanMapObjFun, CategoryTheory.CategoryStruct.comp, CategoryTheory.Iso.hom, Eq, CategoryTheory.NatTrans.app, CategoryTheory.GradedObject.cofanMapObjComp, CategoryTheory.Limits.Cocone.pt