awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
simp :
  Eq (α := (G.obj c₁.pt).obj c₂.pt ⟶ (G.obj (K₁.obj j₁)).obj c₂.pt)
    ((P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } :
      Cone.pt.{v_2, v_5, u_2, u_5} (J := J₂) (F := (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
          { pt := (G.obj c₁.pt).obj c₂.pt,
            π :=
              { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
                naturality := ⋯ :
                NatTrans (obj (const J₂) ((G.obj c₁.pt).obj c₂.pt))
                  ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } :
            Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } ⟶
        ((G.obj (K₁.obj j₁)).mapCone c₂).pt)
    ((G.map (c₁.π.app j₁)).app c₂.pt)
⊢ (P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } ≫
      𝟙 ((G.obj (K₁.obj j₁)).obj c₂.pt) =
    𝟙 ((G.obj c₁.pt).obj c₂.pt) ≫ (G.map (c₁.π.app j₁)).app c₂.pt

Tactic:
simp only [id_comp, comp_id]

Premises used: CategoryTheory.Category.comp_id, CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
refine_2 :
  Eq (α := (G.obj c₁.pt).obj c₂.pt ⟶ (G.obj (K₁.obj j₁)).obj c₂.pt)
    ((P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } :
      Cone.pt.{v_2, v_5, u_2, u_5} (J := J₂) (F := (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
          { pt := (G.obj c₁.pt).obj c₂.pt,
            π :=
              { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
                naturality := ⋯ :
                NatTrans (obj (const J₂) ((G.obj c₁.pt).obj c₂.pt))
                  ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } :
            Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } ⟶
        ((G.obj (K₁.obj j₁)).mapCone c₂).pt)
    ((G.map (c₁.π.app j₁)).app c₂.pt)
refine_1 : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
⊢ Eq (α := (G.obj c₁.pt).obj c₂.pt ⟶ (G.obj (K₁.obj j₁)).obj c₂.pt)
    ((P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } :
      Cone.pt.{v_2, v_5, u_2, u_5} (J := J₂) (F := (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁)
          { pt := (G.obj c₁.pt).obj c₂.pt,
            π :=
              { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
                naturality := ⋯ :
                NatTrans (obj (const J₂) ((G.obj c₁.pt).obj c₂.pt))
                  ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } :
            Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) } ⟶
        ((G.obj (K₁.obj j₁)).mapCone c₂).pt)
    ((G.map (c₁.π.app j₁)).app c₂.pt)

Tactic:
let s : Cone (whiskeringLeft₂ C |>.obj K₁ |>.obj K₂ |>.obj G |>.obj j₁) := ?_
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
s : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    π :=
      { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
        naturality := ⋯ } }
refine_2 : (G.map (c₁.π.app j₁)).app c₂.pt = (P j₁).lift s
⊢ (P j₁).lift s = (G.map (c₁.π.app j₁)).app c₂.pt

Tactic:
symm
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
s : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    π :=
      { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
        naturality := ⋯ } }
refine_2 : ∀ (j : J₂), (G.map (c₁.π.app j₁)).app c₂.pt ≫ (Q₀.obj j₁).π.app j = (P j₁).lift s ≫ (Q₀.obj j₁).π.app j
⊢ (G.map (c₁.π.app j₁)).app c₂.pt = (P j₁).lift s

Tactic:
apply (P j₁).hom_ext

Premises used: CategoryTheory.Limits.IsLimit.hom_ext
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
s : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    π :=
      { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
        naturality := ⋯ } }
j₂ : J₂
⊢ (P j₁).lift s ≫ (Q₀.obj j₁).π.app j₂ = s.π.app j₂

Tactic:
haveI := (P j₁).fac s j₂
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesLimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesLimit K₂ (G.obj x)
c₁ : Cone K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
Q₀ : DiagramOfCones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsLimit (Q₀.obj j₁) := fun (x : J₁) ↦ isLimitOfPreserves (G.obj (K₁.obj x)) hc₂
E₀ : Q₀.conePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.conePoints.obj x)) ⋯
j₁ : J₁
s : Cone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    π :=
      { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
        naturality := ⋯ } }
j₂ : J₂
this :
  (P j₁).lift
        { pt := (G.obj c₁.pt).obj c₂.pt,
          π :=
            { app := fun (k : J₂) ↦ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app k),
              naturality := ⋯ } } ≫
      (G.obj (K₁.obj j₁)).map (c₂.π.app j₂) =
    (G.map (c₁.π.app j₁)).app c₂.pt ≫ (G.obj (K₁.obj j₁)).map (c₂.π.app j₂)
⊢ (G.map (c₁.π.app j₁)).app c₂.pt ≫ (Q₀.obj j₁).π.app j₂ = (P j₁).lift s ≫ (Q₀.obj j₁).π.app j₂

Tactic:
simp only [whiskeringLeft₂_obj_obj_obj_obj_obj,
            Functor.mapCone_pt, Functor.mapCone_π_app, this, Q₀, s]

Premises used: CategoryTheory.Functor.whiskeringLeft₂_obj_obj_obj_obj_obj, CategoryTheory.Functor.mapCone_pt, CategoryTheory.Functor.mapCone_π_app
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
val : IsLimit (G.flip.mapCone₂ c₁ c₂)
⊢ Nonempty (IsLimit (G.flip.mapCone₂ c₁ c₂))

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
val : IsLimit ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))
⊢ IsLimit (G.flip.mapCone₂ c₁ c₂)

Tactic:
refine IsLimit.postcomposeHomEquiv E₁ _ ?_

Premises used: CategoryTheory.Limits.IsLimit.postcomposeHomEquiv, DFunLike.coe
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
val :
  IsLimit (F :=
    Equivalence.functor (Prod.braiding J₁ J₂) ⋙
      Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
    (Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂)))
⊢ IsLimit ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))

Tactic:
apply IsLimit.ofWhiskerEquivalence (e := Prod.braiding _ _)

Premises used: CategoryTheory.Prod.braiding, CategoryTheory.Limits.IsLimit.ofWhiskerEquivalence
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
val :
  (Cone.postcompose (Iso.hom (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
      (G.mapCone₂ c₂ c₁) ≅
    Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))
⊢ IsLimit (F :=
    Equivalence.functor (Prod.braiding J₁ J₂) ⋙
      Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
    (Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂)))

Tactic:
refine IsLimit.equivOfNatIsoOfIso (Iso.refl _) (G.mapCone₂ c₂ c₁) _ ?_ |>.toFun <|
      isLimitOfPreserves₂ G hc₂ hc₁

Premises used: CategoryTheory.Functor.mapCone₂, CategoryTheory.Iso.refl, Equiv.toFun, CategoryTheory.Limits.isLimitOfPreserves₂, CategoryTheory.Limits.IsLimit.equivOfNatIsoOfIso
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
x : J₁ × J₂
j₁ : J₁
j₂ : J₂
⊢ Eq (α :=
    obj
        (obj (const (J₁ × J₂))
          ((Cone.postcompose (Iso.hom (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
              (G.mapCone₂ c₂ c₁)).pt)
        (j₁, j₂) ⟶
      (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)).obj (j₁, j₂))
    (((Cone.postcompose (Iso.hom (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
            (G.mapCone₂ c₂ c₁)).π.app
      (j₁, j₂))
    (Iso.hom
        (Iso.refl
          ((Cone.postcompose (Iso.hom (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
              (G.mapCone₂ c₂ c₁)).pt) ≫
      NatTrans.app
        (Cone.π
          (Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
            ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))))
        (j₁, j₂))

Tactic:
simp [E₁, E₀]
-- Module: Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
Goal:
J₁ : Type u_1
J₂ : Type u_2
inst : Category.{v_1, u_1} J₁
inst_1 : Category.{v_2, u_2} J₂
C₁ : Type u_3
C₂ : Type u_4
C : Type u_5
inst_2 : Category.{v_3, u_3} C₁
inst_3 : Category.{v_4, u_4} C₂
inst_4 : Category.{v_5, u_5} C
K₁ : Functor.{v_1, v_3, u_1, u_3} J₁ C₁
K₂ : Functor.{v_2, v_4, u_2, u_4} J₂ C₂
G : C₁ ⥤ Functor.{v_4, v_5, u_4, u_5} C₂ C
inst_5 : PreservesLimit₂ K₁ K₂ G
c₁ : Cone K₂
hc₁ : IsLimit c₁
c₂ : Cone K₁
hc₂ : IsLimit c₂
E₀ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).flip :=
  Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip))
E₁ : obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₂) K₁) G.flip) ≅
  Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  E₀ ≪≫ (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).uncurryObjFlip
⊢ (Cone.postcompose (Iso.hom (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
      (G.mapCone₂ c₂ c₁) ≅
    Cone.whisker (Equivalence.functor (Prod.braiding J₁ J₂)) ((Cone.postcompose E₁.hom).obj (G.flip.mapCone₂ c₁ c₂))

Tactic:
exact Cone.ext (Iso.refl _) (fun ⟨j₁, j₂⟩ ↦ by simp [E₁, E₀])

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Limits.Cone.ext
-- Module: Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel
Goal:
C₁ : Type u_1
C₂ : Type u_2
C : Type u_3
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} C
inst_3 : HasZeroMorphisms.{v_1, u_1} C₁
inst_4 : HasZeroMorphisms.{v_2, u_2} C₂
inst_5 : HasZeroMorphisms.{v_3, u_3} C
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
c₁ : CokernelCofork f₁
hc₁ : IsColimit c₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
c₂ : CokernelCofork f₂
hc₂ : IsColimit c₂
F : C₁ ⥤ Functor.{v_2, v_3, u_2, u_3} C₂ C
inst_6 : (F.obj c₁.pt).PreservesZeroMorphisms
inst_7 : F.PreservesZeroMorphisms
inst_8 : HasBinaryCoproduct.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)
h₁ :
  CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inl
      (coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂) ≫
        (F.map (Cofork.π c₁)).app Y₂ ≫ (F.obj c₁.pt).map (Cofork.π c₂)) =
    CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inl 0
h₂ :
  CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inr
      (coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂) ≫
        (F.map (Cofork.π c₁)).app Y₂ ≫ (F.obj c₁.pt).map (Cofork.π c₂)) =
    CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inr 0
⊢ Eq (α := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂) ⟶ (F.obj c₁.pt).obj c₂.pt)
    (coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂) ≫
      (F.map (Cofork.π c₁)).app Y₂ ≫ (F.obj c₁.pt).map (Cofork.π c₂))
    0

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel
Goal:
C₁ : Type u_1
C₂ : Type u_2
C : Type u_3
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} C
inst_3 : HasZeroMorphisms.{v_1, u_1} C₁
inst_4 : HasZeroMorphisms.{v_2, u_2} C₂
inst_5 : HasZeroMorphisms.{v_3, u_3} C
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
c₁ : CokernelCofork f₁
hc₁ : IsColimit c₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
c₂ : CokernelCofork f₂
hc₂ : IsColimit c₂
F : C₁ ⥤ Functor.{v_2, v_3, u_2, u_3} C₂ C
inst_6 : (F.obj c₁.pt).PreservesZeroMorphisms
inst_7 : F.PreservesZeroMorphisms
inst_8 : HasBinaryCoproduct.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)
⊢ CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inl
      (coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂) ≫
        (F.map (Cofork.π c₁)).app Y₂ ≫ (F.obj c₁.pt).map (Cofork.π c₂)) =
    CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inl 0

Tactic:
simp [← NatTrans.comp_app_assoc, ← Functor.map_comp]

Premises used: CategoryTheory.NatTrans.comp_app_assoc, CategoryTheory.Functor.map_comp
-- Module: Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel
Goal:
C₁ : Type u_1
C₂ : Type u_2
C : Type u_3
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} C
inst_3 : HasZeroMorphisms.{v_1, u_1} C₁
inst_4 : HasZeroMorphisms.{v_2, u_2} C₂
inst_5 : HasZeroMorphisms.{v_3, u_3} C
X₁ Y₁ : C₁
f₁ : X₁ ⟶ Y₁
c₁ : CokernelCofork f₁
hc₁ : IsColimit c₁
X₂ Y₂ : C₂
f₂ : X₂ ⟶ Y₂
c₂ : CokernelCofork f₂
hc₂ : IsColimit c₂
F : C₁ ⥤ Functor.{v_2, v_3, u_2, u_3} C₂ C
inst_6 : (F.obj c₁.pt).PreservesZeroMorphisms
inst_7 : F.PreservesZeroMorphisms
inst_8 : HasBinaryCoproduct.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)
⊢ CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inr
      (coprod.desc ((F.map f₁).app Y₂) ((F.obj Y₁).map f₂) ≫
        (F.map (Cofork.π c₁)).app Y₂ ≫ (F.obj c₁.pt).map (Cofork.π c₂)) =
    CategoryStruct.comp (Y := coprod.{v_3, u_3} ((F.obj X₁).obj Y₂) ((F.obj Y₁).obj X₂)) coprod.inr 0

Tactic:
simp [← Functor.map_comp]

Premises used: CategoryTheory.Functor.map_comp