theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ := by
  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.
  let α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₁]) }
  let α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
    { app := fun X => eqToHom (by simp [← h₂]) }
  let c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
    (Cones.postcompose (whiskerRight α₁ (pre s.pt K F) : _)).obj (c.toStructuredArrowCone F f₁)
  let c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
    (Cones.postcompose (whiskerRight α₂ (pre s.pt K F) : _)).obj (c.toStructuredArrowCone F f₂)
  -- The two cones can then be combined and we may obtain a cone over the two cones since
  -- `StructuredArrow s.pt F` is cofiltered.
  let c₀ := IsCofiltered.cone (biconeMk _ c₁ c₂)
  let g₁ : c₀.pt ⟶ c₁.pt := c₀.π.app Bicone.left
  let g₂ : c₀.pt ⟶ c₂.pt := c₀.π.app Bicone.right
  -- Then `g₁.right` and `g₂.right` are two maps from the same cone into the `c`.
  have : ∀ j : J, g₁.right ≫ c.π.app j = g₂.right ≫ c.π.app j := by
    intro j
    injection c₀.π.naturality (BiconeHom.left j) with _ e₁
    injection c₀.π.naturality (BiconeHom.right j) with _ e₂
    convert e₁.symm.trans e₂ <;> simp
  have : c.extend g₁.right = c.extend g₂.right := by
    unfold Cone.extend
    congr 1
    ext x
    apply this
  -- And thus they are equal as `c` is the limit.
  have : g₁.right = g₂.right
  calc
    g₁.right = hc.lift (c.extend g₁.right) := by
      

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₁, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K✝ : J ⥤ C
F : C ⥤ D
inst✝ : RepresentablyFlat F
c✝ : Cone K✝
hc✝ : IsLimit c✝
s✝ : Cone (K✝ ⋙ F)
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
α₁ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₁ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₁ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
α₂ : Cone.toStructuredArrow (F.mapCone c) ⋙ map f₂ ⟶ Cone.toStructuredArrow s :=
  NatTrans.mk fun X => eqToHom (_ : StructuredArrow.mk (f₂ ≫ F.map (c.π.app X)) = StructuredArrow.mk (s.π.app X))
c₁ : Cone (Cone.toStructuredArrow s ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (Cone.toStructuredArrowCone c F f₁)
c₂ : Cone (Cone.toStructuredArrow s ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (Cone.toStructuredArrowCone c F f₂)
c₀ : Cone (biconeMk J c₁ c₂) := IsCofiltered.cone (biconeMk J c₁ c₂)
g₁ : c₀.pt ⟶ c₁.pt := c₀.π.app Bicone.left
g₂ : c₀.pt ⟶ c₂.pt := c₀.π.app Bicone.right
this✝ : ∀ (j : J), g₁.right ≫ c.π.app j = g₂.right ≫ c.π.app j
this : Cone.extend c g₁.right = Cone.extend c g₂.right
⊢ g₁.right = IsLimit.lift hc (Cone.extend c g₁.right)

Next tactic:
apply hc.uniq (c.extend _)