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 : PreservesColimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesColimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesColimit K₂ (G.obj x)
c₁ : Cocone K₁
hc₁ : IsColimit c₁
c₂ : Cocone K₂
hc₂ : IsColimit c₂
Q₀ : DiagramOfCocones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCocone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsColimit (Q₀.obj j₁) := fun (j : J₁) ↦ isColimitOfPreserves (G.obj (K₁.obj j)) hc₂
E₀ : Q₀.coconePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.coconePoints.obj x)) ⋯
j₁ : J₁
s : Cocone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    ι :=
      { app := fun (k : J₂) ↦ (G.map (c₁.ι.app j₁)).app (K₂.obj k) ≫ (G.obj c₁.pt).map (c₂.ι.app k), naturality := ⋯ } }
refine_2 : (G.map (c₁.ι.app j₁)).app c₂.pt = (P j₁).desc s
⊢ (P j₁).desc s = (G.map (c₁.ι.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 : PreservesColimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesColimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesColimit K₂ (G.obj x)
c₁ : Cocone K₁
hc₁ : IsColimit c₁
c₂ : Cocone K₂
hc₂ : IsColimit c₂
Q₀ : DiagramOfCocones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCocone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsColimit (Q₀.obj j₁) := fun (j : J₁) ↦ isColimitOfPreserves (G.obj (K₁.obj j)) hc₂
E₀ : Q₀.coconePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.coconePoints.obj x)) ⋯
j₁ : J₁
s : Cocone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    ι :=
      { app := fun (k : J₂) ↦ (G.map (c₁.ι.app j₁)).app (K₂.obj k) ≫ (G.obj c₁.pt).map (c₂.ι.app k), naturality := ⋯ } }
refine_2 : ∀ (j : J₂), (Q₀.obj j₁).ι.app j ≫ (G.map (c₁.ι.app j₁)).app c₂.pt = (Q₀.obj j₁).ι.app j ≫ (P j₁).desc s
⊢ (G.map (c₁.ι.app j₁)).app c₂.pt = (P j₁).desc s

Tactic:
apply (P j₁).hom_ext

Premises used: CategoryTheory.Limits.IsColimit.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 : PreservesColimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesColimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesColimit K₂ (G.obj x)
c₁ : Cocone K₁
hc₁ : IsColimit c₁
c₂ : Cocone K₂
hc₂ : IsColimit c₂
Q₀ : DiagramOfCocones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCocone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsColimit (Q₀.obj j₁) := fun (j : J₁) ↦ isColimitOfPreserves (G.obj (K₁.obj j)) hc₂
E₀ : Q₀.coconePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.coconePoints.obj x)) ⋯
j₁ : J₁
s : Cocone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    ι :=
      { app := fun (k : J₂) ↦ (G.map (c₁.ι.app j₁)).app (K₂.obj k) ≫ (G.obj c₁.pt).map (c₂.ι.app k), naturality := ⋯ } }
j₂ : J₂
⊢ (Q₀.obj j₁).ι.app j₂ ≫ (P j₁).desc s = s.ι.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 : PreservesColimit₂ K₁ K₂ G
inst_6 : ∀ (x : C₂), PreservesColimit K₁ (G.flip.obj x)
inst_7 : ∀ (x : C₁), PreservesColimit K₂ (G.obj x)
c₁ : Cocone K₁
hc₁ : IsColimit c₁
c₂ : Cocone K₂
hc₂ : IsColimit c₂
Q₀ : DiagramOfCocones (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G) :=
  { obj := fun (j₁ : J₁) ↦ (G.obj (K₁.obj j₁)).mapCocone c₂,
    map := fun {j j' : J₁} (f : j ⟶ j') ↦ { hom := (G.map (K₁.map f)).app c₂.pt, w := ⋯ }, id := ⋯, comp := ⋯ }
P : (j₁ : J₁) → IsColimit (Q₀.obj j₁) := fun (j : J₁) ↦ isColimitOfPreserves (G.obj (K₁.obj j)) hc₂
E₀ : Q₀.coconePoints ≅ K₁ ⋙ G.flip.obj c₂.pt := NatIso.ofComponents (fun (x : J₁) ↦ Iso.refl (Q₀.coconePoints.obj x)) ⋯
j₁ : J₁
s : Cocone ((obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G).obj j₁) :=
  { pt := (G.obj c₁.pt).obj c₂.pt,
    ι :=
      { app := fun (k : J₂) ↦ (G.map (c₁.ι.app j₁)).app (K₂.obj k) ≫ (G.obj c₁.pt).map (c₂.ι.app k), naturality := ⋯ } }
j₂ : J₂
this :
  (G.obj (K₁.obj j₁)).map (c₂.ι.app j₂) ≫
      (P j₁).desc
        { pt := (G.obj c₁.pt).obj c₂.pt,
          ι :=
            { app := fun (k : J₂) ↦ (G.map (c₁.ι.app j₁)).app (K₂.obj k) ≫ (G.obj c₁.pt).map (c₂.ι.app k),
              naturality := ⋯ } } =
    (G.map (c₁.ι.app j₁)).app (K₂.obj j₂) ≫ (G.obj c₁.pt).map (c₂.ι.app j₂)
⊢ (Q₀.obj j₁).ι.app j₂ ≫ (G.map (c₁.ι.app j₁)).app c₂.pt = (Q₀.obj j₁).ι.app j₂ ≫ (P j₁).desc s

Tactic:
simp only [Functor.mapCocone_pt,
            Functor.const_obj_obj, Functor.mapCocone_ι_app, NatTrans.naturality, this, Q₀, s]

Premises used: CategoryTheory.Functor.mapCocone_ι_app, CategoryTheory.NatTrans.naturality, CategoryTheory.Functor.const_obj_obj, CategoryTheory.Functor.mapCocone_pt
-- 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 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit c₂
val : IsColimit (G.flip.mapCocone₂ c₁ c₂)
⊢ Nonempty (IsColimit (G.flip.mapCocone₂ 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 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit 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 : IsColimit ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂))
⊢ IsColimit (G.flip.mapCocone₂ c₁ c₂)

Tactic:
refine IsColimit.precomposeInvEquiv E₁ _ ?_

Premises used: DFunLike.coe, CategoryTheory.Limits.IsColimit.precomposeInvEquiv
-- 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 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit 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 :
  IsColimit (F :=
    Equivalence.functor (Prod.braiding J₁ J₂) ⋙
      Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
    (Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
      ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂)))
⊢ IsColimit ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂))

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

Premises used: CategoryTheory.Prod.braiding, CategoryTheory.Limits.IsColimit.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 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit 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 :
  (Cocone.precompose (Iso.inv (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
      (G.mapCocone₂ c₂ c₁) ≅
    Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
      ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂))
⊢ IsColimit (F :=
    Equivalence.functor (Prod.braiding J₁ J₂) ⋙
      Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
    (Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
      ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂)))

Tactic:
refine IsColimit.equivOfNatIsoOfIso (Iso.refl _) (G.mapCocone₂ c₂ c₁) _ ?_ |>.toFun <|
      isColimitOfPreserves₂ G hc₂ hc₁

Premises used: CategoryTheory.Limits.isColimitOfPreserves₂, CategoryTheory.Limits.IsColimit.equivOfNatIsoOfIso, CategoryTheory.Iso.refl, Equiv.toFun, CategoryTheory.Functor.mapCocone₂
-- 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 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit 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 uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)).obj (j₁, j₂) ⟶
      Cocone.pt (F := obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
        (Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
            ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂)) :
          Cocone
            (Equivalence.functor (Prod.braiding J₁ J₂) ⋙
              Prod.swap J₂ J₁ ⋙ obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))
    (((Cocone.precompose (Iso.inv (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
              (G.mapCocone₂ c₂ c₁)).ι.app
        (j₁, j₂) ≫
      Iso.hom
        (Iso.refl
          ((Cocone.precompose (Iso.inv (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
              (G.mapCocone₂ c₂ c₁)).pt))
    (NatTrans.app
      (Cocone.ι
        (Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
          ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ 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 : PreservesColimit₂ K₁ K₂ G
c₁ : Cocone K₂
hc₁ : IsColimit c₁
c₂ : Cocone K₁
hc₂ : IsColimit 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
⊢ (Cocone.precompose (Iso.inv (Iso.refl (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))))).obj
      (G.mapCocone₂ c₂ c₁) ≅
    Cocone.whisker (Equivalence.functor (Prod.braiding J₁ J₂))
      ((Cocone.precompose E₁.inv).obj (G.flip.mapCocone₂ c₁ c₂))

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

Premises used: CategoryTheory.Limits.Cocone.ext, CategoryTheory.Iso.refl
-- 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 K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
c₃ : Cone (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
hc₃ : IsLimit c₃
j : J₁ × J₂
dsimp :
  ((isLimitOfPreserves₂ G hc₁ hc₂).conePointUniqueUpToIso hc₃).hom ≫ c₃.π.app j =
    NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2)
⊢ (isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).hom ≫ c₃.π.app j =
    NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2)

Tactic:
dsimp [isoObjConePointsOfIsLimit, Functor.mapCocone₂]

Premises used: CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit, CategoryTheory.Functor.mapCocone₂
-- 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 K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
c₃ : Cone (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
hc₃ : IsLimit c₃
j : J₁ × J₂
rw :
  NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2) =
    (isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).hom ≫ c₃.π.app j
⊢ (isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).inv ≫
      NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2) =
    c₃.π.app j

Tactic:
rw [Iso.inv_comp_eq]

Premises used: CategoryTheory.Iso.inv_comp_eq
-- 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 K₁
hc₁ : IsLimit c₁
c₂ : Cone K₂
hc₂ : IsLimit c₂
c₃ : Cone (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G))
hc₃ : IsLimit c₃
j : J₁ × J₂
⊢ NatTrans.app (G.map (c₁.π.app j.1)) c₂.pt ≫ (G.obj (K₁.obj j.1)).map (c₂.π.app j.2) =
    (isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).hom ≫ c₃.π.app j

Tactic:
simp
-- 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 : HasLimit K₁
inst_7 : HasLimit K₂
j : J₁ × J₂
⊢ Eq (α := (G.obj (limit K₁)).obj (limit K₂) ⟶ (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)).obj j)
    (CategoryStruct.comp (Y := limit (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)))
      (Iso.inv (isoLimitUncurryWhiskeringLeft₂ K₁ K₂ G) :
        (G.obj (limit K₁)).obj (limit K₂) ⟶ limit (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)))
      (limit.π (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)) j :
        limit (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)) ⟶
          (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)).obj j))
    (NatTrans.app (G.map (limit.π K₁ j.1)) (limit K₂) ≫ (G.obj (K₁.obj j.1)).map (limit.π K₂ j.2))

Tactic:
isoObjConePointsOfIsLimit_hom_comp_π G
    (limit.isLimit _) (limit.isLimit _) (limit.isLimit _) _

Premises used: CategoryTheory.Limits.limit.π, CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit_hom_comp_π, CategoryTheory.Limits.limit.isLimit, CategoryTheory.Limits.limit
-- 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 : HasLimit K₁
inst_7 : HasLimit K₂
j : J₁ × J₂
⊢ Eq (α := limit (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)) ⟶ (G.obj (K₁.obj j.1)).obj (K₂.obj j.2))
    (CategoryStruct.comp (Y := (G.obj (limit K₁)).obj (limit K₂)) (Iso.hom (isoLimitUncurryWhiskeringLeft₂ K₁ K₂ G))
        (NatTrans.app (G.map (limit.π K₁ j.1)) (limit K₂) ≫ (G.obj (K₁.obj j.1)).map (limit.π K₂ j.2)) :
      limit (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)) ⟶ (G.obj (K₁.obj j.1)).obj (K₂.obj j.2))
    (limit.π (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)) j :
      limit (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)) ⟶
        (obj uncurry (obj (obj (obj (whiskeringLeft₂ C) K₁) K₂) G)).obj j)

Tactic:
isoObjConePointsOfIsColimit_inv_comp_map_π G
    (limit.isLimit _) (limit.isLimit _) (limit.isLimit _) _

Premises used: CategoryTheory.Limits.limit.π, CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsColimit_inv_comp_map_π, CategoryTheory.Limits.limit.isLimit, CategoryTheory.Limits.limit
-- 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
dsimp :
  (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
⊢ ((Cone.postcompose E₀.hom).obj (coneOfConeUncurry P (G.mapCone₂ c₁ c₂))).π.app j₁ =
    Iso.hom (Iso.refl ((Cone.postcompose E₀.hom).obj (coneOfConeUncurry P (G.mapCone₂ c₁ c₂))).pt) ≫
      ((G.flip.obj c₂.pt).mapCone c₁).π.app j₁

Tactic:
dsimp [E₀, Q₀]