l3lab/ntp-mathlib huggingface dataset:
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    

Proof state:
case this.cone_objs
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
⊢ ∃ W x x, True

Next tactic:
let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    

Proof state:
case this.cone_objs
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
⊢ ∃ W x x, True

Next tactic:
let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    

Proof state:
case this.cone_objs
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
W' : StructuredArrow W.right F := IsCofiltered.min (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
⊢ ∃ W x x, True

Next tactic:
let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    

Proof state:
case this.cone_objs
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
W' : StructuredArrow W.right F := IsCofiltered.min (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Y'' : W' ⟶ StructuredArrow.mk Y'.right :=
  IsCofiltered.minToLeft (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
⊢ ∃ W x x, True

Next tactic:
let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    

Proof state:
case this.cone_objs
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
W' : StructuredArrow W.right F := IsCofiltered.min (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Y'' : W' ⟶ StructuredArrow.mk Y'.right :=
  IsCofiltered.minToLeft (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Z'' : W' ⟶ StructuredArrow.mk Z'.right :=
  IsCofiltered.minToRight (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
⊢ ∃ W x x, True

Next tactic:
use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    

Proof state:
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
W' : StructuredArrow W.right F := IsCofiltered.min (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Y'' : W' ⟶ StructuredArrow.mk Y'.right :=
  IsCofiltered.minToLeft (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Z'' : W' ⟶ StructuredArrow.mk Z'.right :=
  IsCofiltered.minToRight (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
⊢ ∃ x x, True

Next tactic:
use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by 

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
W' : StructuredArrow W.right F := IsCofiltered.min (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Y'' : W' ⟶ StructuredArrow.mk Y'.right :=
  IsCofiltered.minToLeft (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Z'' : W' ⟶ StructuredArrow.mk Z'.right :=
  IsCofiltered.minToRight (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
⊢ (StructuredArrow.mk (W.hom ≫ G.map W'.hom)).hom ≫ (F ⋙ G).map Y''.right = Y.hom

Next tactic:
simp [← G.map_comp]
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    

Proof state:
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
W' : StructuredArrow W.right F := IsCofiltered.min (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Y'' : W' ⟶ StructuredArrow.mk Y'.right :=
  IsCofiltered.minToLeft (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Z'' : W' ⟶ StructuredArrow.mk Z'.right :=
  IsCofiltered.minToRight (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
⊢ ∃ x, True

Next tactic:
use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by 

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
W : StructuredArrow X G := IsCofiltered.min (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Y' : W ⟶ StructuredArrow.mk Y.hom := IsCofiltered.minToLeft (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
Z' : W ⟶ StructuredArrow.mk Z.hom := IsCofiltered.minToRight (StructuredArrow.mk Y.hom) (StructuredArrow.mk Z.hom)
W' : StructuredArrow W.right F := IsCofiltered.min (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Y'' : W' ⟶ StructuredArrow.mk Y'.right :=
  IsCofiltered.minToLeft (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
Z'' : W' ⟶ StructuredArrow.mk Z'.right :=
  IsCofiltered.minToRight (StructuredArrow.mk Y'.right) (StructuredArrow.mk Z'.right)
⊢ (StructuredArrow.mk (W.hom ≫ G.map W'.hom)).hom ≫ (F ⋙ G).map Z''.right = Z.hom

Next tactic:
simp [← G.map_comp]
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
  · 

Proof state:
case this.cone_maps
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
⊢ ∀ ⦃X_1 Y : StructuredArrow X (F ⋙ G)⦄ (f g : X_1 ⟶ Y), ∃ W h, h ≫ f = h ≫ g

Next tactic:
intro Y Z f g
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
  · intro Y Z f g
    

Proof state:
case this.cone_maps
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
f g : Y ⟶ Z
⊢ ∃ W h, h ≫ f = h ≫ g

Next tactic:
let W :=
      @IsCofiltered.eq (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom) (StructuredArrow.homMk (F.map f.right) (StructuredArrow.w f))
        (StructuredArrow.homMk (F.map g.right) (StructuredArrow.w g))
instance RepresentablyFlat.comp (F : C ⥤ D) (G : D ⥤ E) [RepresentablyFlat F]
    [RepresentablyFlat G] : RepresentablyFlat (F ⋙ G) := by
  constructor
  intro X
  have : Nonempty (StructuredArrow X (F ⋙ G)) := by
    have f₁ : StructuredArrow X G := Nonempty.some inferInstance
    have f₂ : StructuredArrow f₁.right F := Nonempty.some inferInstance
    exact ⟨StructuredArrow.mk (f₁.hom ≫ G.map f₂.hom)⟩
  suffices : IsCofilteredOrEmpty (StructuredArrow X (F ⋙ G))
  · constructor
  constructor
  · intro Y Z
    let W :=
      @IsCofiltered.min (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom)
    let Y' : W ⟶ _ := IsCofiltered.minToLeft _ _
    let Z' : W ⟶ _ := IsCofiltered.minToRight _ _
    let W' :=
      @IsCofiltered.min (StructuredArrow W.right F) _ _ (StructuredArrow.mk Y'.right)
        (StructuredArrow.mk Z'.right)
    let Y'' : W' ⟶ _ := IsCofiltered.minToLeft _ _
    let Z'' : W' ⟶ _ := IsCofiltered.minToRight _ _
    use StructuredArrow.mk (W.hom ≫ G.map W'.hom)
    use StructuredArrow.homMk Y''.right (by simp [← G.map_comp])
    use StructuredArrow.homMk Z''.right (by simp [← G.map_comp])
  · intro Y Z f g
    let W :=
      @IsCofiltered.eq (StructuredArrow X G) _ _ (StructuredArrow.mk Y.hom)
        (StructuredArrow.mk Z.hom) (StructuredArrow.homMk (F.map f.right) (StructuredArrow.w f))
        (StructuredArrow.homMk (F.map g.right) (StructuredArrow.w g))
    

Proof state:
case this.cone_maps
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : RepresentablyFlat F
inst✝ : RepresentablyFlat G
X : E
this : Nonempty (StructuredArrow X (F ⋙ G))
Y Z : StructuredArrow X (F ⋙ G)
f g : Y ⟶ Z
W : StructuredArrow X G :=
  IsCofiltered.eq (StructuredArrow.homMk (F.map f.right)) (StructuredArrow.homMk (F.map g.right))
⊢ ∃ W h, h ≫ f = h ≫ g

Next tactic:
let h : W ⟶ _ := IsCofiltered.eqHom _ _