-- 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_1 : (i₁, i₂₃) ∈ ρ₂₃.q ⁻¹' {j}
x_1 :
  Discrete
    (↑(p' ⁻¹'
          {((↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi_1) :
                  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_1) :
                  I₁ × ρ₂₃.I₂₃).2)}) :
      Type (max (max u_7 u_8) u_9))
i₁' : I₁
i₂ : I₂
i₃ : I₃
hi :
  (i₁', i₂, i₃) ∈
    p' ⁻¹'
      {((↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi_1) :
              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_1) :
              I₁ × ρ₂₃.I₂₃).2)}
obtain :
  ∀ (hi : (i₁', i₂₃) ∈ ρ₂₃.q ⁻¹' {j})
    (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)))
    (hi_2 :
      (i₁', i₂, i₃) ∈
        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)}),
    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)))
        {
          as :=
            Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦
              x ∈
                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)})
              (i₁', i₂, i₃) hi_2 :
          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)) } =
      ((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
        {
          as :=
            Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦
              x ∈
                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)})
              (i₁', i₂, i₃) hi_2 }
⊢ 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_1) :
                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_1) :
                I₁ × ρ₂₃.I₂₃).2)))
      {
        as :=
          Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦
            x ∈
              p' ⁻¹'
                {((↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi_1) :
                        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_1) :
                        I₁ × ρ₂₃.I₂₃).2)})
            (i₁', i₂, i₃) hi :
        Discrete
          (↑(p' ⁻¹'
                {((↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi_1) :
                        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_1) :
                        I₁ × ρ₂₃.I₂₃).2)}) :
            Type (max (max u_7 u_8) u_9)) } =
    ((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_1) :
                    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_1) :
                    I₁ × ρ₂₃.I₂₃).1)).obj
      {
        as :=
          Subtype.mk (p := fun (x : I₁ × I₂ × I₃) ↦
            x ∈
              p' ⁻¹'
                {((↑(Subtype.mk (p := fun (x : Prod.{u_7, u_11} I₁ ρ₂₃.I₂₃) ↦ x ∈ ρ₂₃.q ⁻¹' {j}) (i₁, i₂₃) hi_1) :
                        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_1) :
                        I₁ × ρ₂₃.I₂₃).2)})
            (i₁', i₂, i₃) hi }

Tactic:
obtain rfl : i₁' = i₁ := congr_arg _root_.Prod.fst hi

Premises used: Prod.fst, congr_arg, rfl