-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory (goal)
Goal:
C : Type u
inst : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type s
inst_1 : (X Y : C) → FunLike.{u_1 + 1, s + 1, s + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{s, v, u, u_1} C FC (CC := CC)
J : MulticospanShape
I : MulticospanIndex.{w, w', v, u} J C
x :
  { x : (i : J.L) → ToType.{s, v, u, u_1} (FC := FC) (CC := CC) (I.left i) //
    ∀ (i : J.R),
      Eq.{s + 1} (α := CC (I.right i))
        ((hom (FC := FC) (CC := CC) (I.fst i) : (a : CC (I.left (J.fst i))) → CC (I.right i)) (x (J.fst i)) :
          CC (I.right i))
        ((hom (FC := FC) (CC := CC) (I.snd i) : (a : CC (I.left (J.snd i))) → CC (I.right i)) (x (J.snd i)) :
          CC (I.right i)) }
⊢ (fun (x : (↑(I.multicospan ⋙ forget C (FC := FC) (CC := CC)).sections : Type (max (max s w) w'))) ↦
        (Subtype.mk (p := fun (x : (i : J.L) → ToType.{s, v, u, u_1} (FC := FC) (CC := CC) (I.left i)) ↦
            ∀ (i : J.R),
              Eq.{s + 1} (α := CC (I.right i))
                ((hom (FC := FC) (CC := CC) (I.fst i) : (a : CC (I.left (J.fst i))) → CC (I.right i)) (x (J.fst i)) :
                  CC (I.right i))
                ((hom (FC := FC) (CC := CC) (I.snd i) : (a : CC (I.left (J.snd i))) → CC (I.right i)) (x (J.snd i)) :
                  CC (I.right i)))
            (fun (x_1 : J.L) ↦
              (↑x : (j : WalkingMulticospan J) → (I.multicospan ⋙ forget C).obj j) (WalkingMulticospan.left x_1))
            ⋯ :
          { x : (i : J.L) → ToType.{s, v, u, u_1} (FC := FC) (CC := CC) (I.left i) //
            ∀ (i : J.R),
              Eq.{s + 1} (α := CC (I.right i))
                ((hom (FC := FC) (CC := CC) (I.fst i) : (a : CC (I.left (J.fst i))) → CC (I.right i)) (x (J.fst i)) :
                  CC (I.right i))
                ((hom (FC := FC) (CC := CC) (I.snd i) : (a : CC (I.left (J.snd i))) → CC (I.right i)) (x (J.snd i)) :
                  CC (I.right i)) }))
      ((fun
            (x :
              { x : (i : J.L) → ToType.{s, v, u, u_1} (FC := FC) (CC := CC) (I.left i) //
                ∀ (i : J.R),
                  Eq.{s + 1} (α := CC (I.right i))
                    ((hom (FC := FC) (CC := CC) (I.fst i) : (a : CC (I.left (J.fst i))) → CC (I.right i))
                        (x (J.fst i)) :
                      CC (I.right i))
                    ((hom (FC := FC) (CC := CC) (I.snd i) : (a : CC (I.left (J.snd i))) → CC (I.right i))
                        (x (J.snd i)) :
                      CC (I.right i)) }) ↦
          Subtype.mk (p :=
            fun (x : (j : WalkingMulticospan J) → (I.multicospan ⋙ forget C (FC := FC) (CC := CC)).obj j) ↦
            x ∈ (I.multicospan ⋙ forget C (FC := FC) (CC := CC)).sections)
            (fun (j : WalkingMulticospan J) ↦
              match (motive := (j : WalkingMulticospan J) → (I.multicospan ⋙ forget C (FC := FC) (CC := CC)).obj j)
                j with
              | WalkingMulticospan.left a => (↑x : (i : J.L) → ToType (I.left i)) a
              | WalkingMulticospan.right b =>
                (hom (FC := FC) (CC := CC) (I.fst b) : (a : CC (I.left (J.fst b))) → CC (I.right b))
                  ((↑x : (i : J.L) → ToType (I.left i)) (J.fst b)))
            ⋯)
        x) =
    x
have a : ∀ (i : J.L),
  (↑((fun (x : (↑(I.multicospan ⋙ forget C (FC := FC) (CC := CC)).sections : Type (max (max s w) w'))) ↦
              (Subtype.mk (p := fun (x : (i : J.L) → ToType.{s, v, u, u_1} (FC := FC) (CC := CC) (I.left i)) ↦
                  ∀ (i : J.R),
                    Eq.{s + 1} (α := CC (I.right i))
                      ((hom (FC := FC) (CC := CC) (I.fst i) : (a : CC (I.left (J.fst i))) → CC (I.right i))
                          (x (J.fst i)) :
                        CC (I.right i))
                      ((hom (FC := FC) (CC := CC) (I.snd i) : (a : CC (I.left (J.snd i))) → CC (I.right i))
                          (x (J.snd i)) :
                        CC (I.right i)))
                  (fun (x_1 : J.L) ↦
                    (↑x : (j : WalkingMulticospan J) → (I.multicospan ⋙ forget C).obj j) (WalkingMulticospan.left x_1))
                  sorry :
                { x : (i : J.L) → ToType.{s, v, u, u_1} (FC := FC) (CC := CC) (I.left i) //
                  ∀ (i : J.R),
                    Eq.{s + 1} (α := CC (I.right i))
                      ((hom (FC := FC) (CC := CC) (I.fst i) : (a : CC (I.left (J.fst i))) → CC (I.right i))
                          (x (J.fst i)) :
                        CC (I.right i))
                      ((hom (FC := FC) (CC := CC) (I.snd i) : (a : CC (I.left (J.snd i))) → CC (I.right i))
                          (x (J.snd i)) :
                        CC (I.right i)) }))
            ((fun
                  (x :
                    { x : (i : J.L) → ToType.{s, v, u, u_1} (FC := FC) (CC := CC) (I.left i) //
                      ∀ (i : J.R),
                        Eq.{s + 1} (α := CC (I.right i))
                          ((hom (FC := FC) (CC := CC) (I.fst i) : (a : CC (I.left (J.fst i))) → CC (I.right i))
                              (x (J.fst i)) :
                            CC (I.right i))
                          ((hom (FC := FC) (CC := CC) (I.snd i) : (a : CC (I.left (J.snd i))) → CC (I.right i))
                              (x (J.snd i)) :
                            CC (I.right i)) }) ↦
                Subtype.mk (p :=
                  fun (x : (j : WalkingMulticospan J) → (I.multicospan ⋙ forget C (FC := FC) (CC := CC)).obj j) ↦
                  x ∈ (I.multicospan ⋙ forget C (FC := FC) (CC := CC)).sections)
                  (fun (j : WalkingMulticospan J) ↦
                    match (motive :=
                      (j : WalkingMulticospan J) → (I.multicospan ⋙ forget C (FC := FC) (CC := CC)).obj j) j with
                    | WalkingMulticospan.left a => (↑x : (i : J.L) → ToType (I.left i)) a
                    | WalkingMulticospan.right b =>
                      (hom (FC := FC) (CC := CC) (I.fst b) : (a : CC (I.left (J.fst b))) → CC (I.right b))
                        ((↑x : (i : J.L) → ToType (I.left i)) (J.fst b)))
                  sorry)
              x)) :
        (i : J.L) → ToType (I.left i))
      i =
    (↑x : (i : J.L) → ToType (I.left i)) i
Tactic: ext i