awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
j : J
left_eq :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).left
⊢ CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right

Tactic:
injection c₀.π.naturality (BiconeHom.left j) with _ e₁

Premises used: CategoryTheory.Limits.Cone.π, CategoryTheory.NatTrans.naturality, CategoryTheory.BiconeHom.left
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
j : J
left_eq :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).left
e₁ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right
⊢ CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).left

Tactic:
injection c₀.π.naturality (BiconeHom.right j) with _ e₂

Premises used: CategoryTheory.Limits.Cone.π, CategoryTheory.BiconeHom.right, CategoryTheory.NatTrans.naturality
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
j : J
left_eq_1 :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).left
e₁ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right
left_eq :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).left
⊢ CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right

Tactic:
injection c₀.π.naturality (BiconeHom.right j) with _ e₂

Premises used: CategoryTheory.Limits.Cone.π, CategoryTheory.BiconeHom.right, CategoryTheory.NatTrans.naturality
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
j : J
left_eq :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).left
e₁ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right
left_eq_1 :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).left
e₂ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right
⊢ CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right =
    CommaMorphism.right (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right

Tactic:
e₁.symm.trans e₂

Premises used: Eq.symm, Eq.trans
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
j : J
left_eq :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).left
e₁ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right
left_eq_1 :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).left
e₂ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right
h :
  (c₀.pt.right ⟶ K.obj j) =
      ((Functor.obj (C := Bicone J) (Functor.obj (Functor.const (Bicone J)) c₀.pt) Bicone.left).right ⟶
        ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right) →
    c₁.pt.right = ((biconeMk J c₁ c₂).obj Bicone.left).right →
      K.obj j = ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right →
        c.π.app j = ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right
h₃ :
  (c₀.pt.right ⟶ K.obj j) =
      ((Functor.obj (C := Bicone J) (Functor.obj (Functor.const (Bicone J)) c₀.pt) Bicone.left).right ⟶
        ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right) →
    c₂.pt.right = ((biconeMk J c₁ c₂).obj Bicone.right).right →
      K.obj j = ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right →
        c.π.app j = ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right
⊢ Hom.right g₁ ≫ c.π.app j = Hom.right g₂ ≫ c.π.app j

Tactic:
convert! e₁.symm.trans e₂

Premises used: Eq.symm, Eq.trans
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
j : J
left_eq :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).left
e₁ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right
left_eq_1 :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).left
e₂ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right
e_1 :
  (c₀.pt.right ⟶ K.obj j) =
    ((Functor.obj (C := Bicone J) (Functor.obj (Functor.const (Bicone J)) c₀.pt) Bicone.left).right ⟶
      ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right)
e_4 : c₁.pt.right = ((biconeMk J c₁ c₂).obj Bicone.left).right
e_5 : K.obj j = ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right
⊢ c.π.app j = ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right

Tactic:
simp [c₁, c₂]
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
j : J
left_eq :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).left
e₁ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.left j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.left) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.left j)).right
left_eq_1 :
  CommaMorphism.left (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.left (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.left (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).left
e₂ :
  CommaMorphism.right (Functor.map (Functor.obj (Functor.const (Bicone J)) c₀.pt) (BiconeHom.right j)) ≫
      CommaMorphism.right (c₀.π.app (Bicone.diagram j)) =
    CommaMorphism.right (c₀.π.app Bicone.right) ≫ ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right
e_1 :
  (c₀.pt.right ⟶ K.obj j) =
    ((Functor.obj (C := Bicone J) (Functor.obj (Functor.const (Bicone J)) c₀.pt) Bicone.left).right ⟶
      ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right)
e_4 : c₂.pt.right = ((biconeMk J c₁ c₂).obj Bicone.right).right
e_5 : K.obj j = ((biconeMk J c₁ c₂).obj (Bicone.diagram j)).right
⊢ c.π.app j = ((biconeMk J c₁ c₂).map (BiconeHom.right j)).right

Tactic:
simp [c₁, c₂]
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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
⊢ ∀ (j : J), Hom.right g₁ ≫ c.π.app j = Hom.right g₂ ≫ c.π.app j

Tactic:
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 [c₁, c₂]
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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), Hom.right g₁ ≫ c.π.app j = Hom.right g₂ ≫ c.π.app j
e_π : Functor.map (Functor.const J) (Hom.right g₁) ≫ c.π = Functor.map (Functor.const J) (Hom.right g₂) ≫ c.π
⊢ { pt := c₀.pt.right, π := Functor.map (Functor.const J) (Hom.right g₁) ≫ c.π : Cone K } =
    { pt := c₀.pt.right, π := Functor.map (Functor.const J) (Hom.right g₂) ≫ c.π }

Tactic:
congr 1
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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), Hom.right g₁ ≫ c.π.app j = Hom.right g₂ ≫ c.π.app j
e_π :
  ∀ (x : J),
    (Functor.map (Functor.const J) (Hom.right g₁) ≫ c.π).app x =
      (Functor.map (Functor.const J) (Hom.right g₂) ≫ c.π).app x
⊢ Functor.map (Functor.const J) (Hom.right g₁) ≫ c.π = Functor.map (Functor.const J) (Hom.right g₂) ≫ c.π

Tactic:
ext x
-- Module: Mathlib.CategoryTheory.Functor.Flat
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J : Type v₁
inst_2 : SmallCategory J
inst_3 : FinCategory J
F : Functor.{v₁, v₂, u₁, u₂} C D
inst_4 : 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 : J) ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow :=
  { app := fun (X : J) ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.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), Hom.right g₁ ≫ c.π.app j = Hom.right g₂ ≫ c.π.app j
x : J
⊢ (Functor.map (Functor.const J) (Hom.right g₁) ≫ c.π).app x =
    (Functor.map (Functor.const J) (Hom.right g₂) ≫ c.π).app x

Tactic:
apply this