-- Mathlib/CategoryTheory/Functor/Flat.lean
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

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
F : C ⥤ D
inst✝ : RepresentablyFlat 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 : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (c.toStructuredArrowCone 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
⊢ f₁ = f₂

Tactic:
have : c.extend g₁.right = c.extend g₂.right := by
  unfold Cone.extend
  congr 1
  ext x
  apply this

Result:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
F : C ⥤ D
inst✝ : RepresentablyFlat 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 : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (c.toStructuredArrowCone 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 : c.extend g₁.right = c.extend g₂.right
⊢ f₁ = f₂

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
F : C ⥤ D
inst✝ : RepresentablyFlat 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 : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (c.toStructuredArrowCone 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
⊢ c.extend g₁.right = c.extend g₂.right

Tactic:
unfold Cone.extend

Result:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
F : C ⥤ D
inst✝ : RepresentablyFlat 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 : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (c.toStructuredArrowCone 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
⊢ { pt := c₀.pt.right, π := c.extensions.app (op c₀.pt.right) { down := g₁.right } } =
    { pt := c₀.pt.right, π := c.extensions.app (op c₀.pt.right) { down := g₂.right } }

Proof state:
case x
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
F : C ⥤ D
inst✝ : RepresentablyFlat 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 : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (c.toStructuredArrowCone 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 : c.extend g₁.right = c.extend g₂.right
j✝ : J
⊢ g₂.right ≫ c.π.app j✝ = (c.extend g₂.right).π.app j✝

Tactic:
rfl

Result:
no goals