l3lab/ntp-mathlib huggingface dataset:
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
⊢ ((HasLimit.isoOfNatIso
            (NatIso.ofComponents fun k =>
                colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).symm).hom ≫
        (colimitLimitIso (F ⋙ diagramFunctor J D X)).inv ≫
          (HasColimit.isoOfNatIso
              (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))
                (limit.isLimit (F ⋙ diagramFunctor J D X)))).inv) ≫
      (plusMap J (limit.π F k)).app (op X) =
    limit.π (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k

Next tactic:
rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
⊢ (plusMap J (limit.π F k)).app (op X) =
    (HasColimit.isoOfNatIso
          (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))
            (limit.isLimit (F ⋙ diagramFunctor J D X)))).hom ≫
      (colimitLimitIso (F ⋙ diagramFunctor J D X)).hom ≫
        (HasLimit.isoOfNatIso
              (NatIso.ofComponents fun k =>
                  colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).symm).inv ≫
          limit.π (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k

Next tactic:
refine' colimit.hom_ext (fun j => _)
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ colimit.ι (diagram J (limit F) (op X).unop) j ≫ (plusMap J (limit.π F k)).app (op X) =
    colimit.ι (diagram J (limit F) (op X).unop) j ≫
      (HasColimit.isoOfNatIso
            (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))
              (limit.isLimit (F ⋙ diagramFunctor J D X)))).hom ≫
        (colimitLimitIso (F ⋙ diagramFunctor J D X)).hom ≫
          (HasLimit.isoOfNatIso
                (NatIso.ofComponents fun k =>
                    colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).symm).inv ≫
            limit.π (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k

Next tactic:
dsimp [plusMap]
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  dsimp [plusMap]
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ colimit.ι (diagram J (limit F) X) j ≫ colimMap (diagramNatTrans J (limit.π F k) X) =
    colimit.ι (diagram J (limit F) X) j ≫
      (HasColimit.isoOfNatIso
            (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))
              (limit.isLimit (F ⋙ diagramFunctor J D X)))).hom ≫
        (colimitLimitIso (F ⋙ diagramFunctor J D X)).hom ≫
          (HasLimit.isoOfNatIso
                (NatIso.ofComponents fun k =>
                    colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).symm).inv ≫
            limit.π (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k

Next tactic:
simp only [HasColimit.isoOfNatIso_ι_hom_assoc, ι_colimMap]
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  dsimp [plusMap]
  simp only [HasColimit.isoOfNatIso_ι_hom_assoc, ι_colimMap]
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ (diagramNatTrans J (limit.π F k) X).app j ≫ colimit.ι (diagram J (F.obj k) X) j =
    (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (diagramFunctor J D X) (limit.isLimit F))
              (limit.isLimit (F ⋙ diagramFunctor J D X))).hom.app
        j ≫
      colimit.ι (limit (F ⋙ diagramFunctor J D X)) j ≫
        (colimitLimitIso (F ⋙ diagramFunctor J D X)).hom ≫
          (HasLimit.isoOfNatIso
                (NatIso.ofComponents fun k =>
                    colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).symm).inv ≫
            limit.π (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k

Next tactic:
dsimp [IsLimit.conePointUniqueUpToIso, HasLimit.isoOfNatIso, IsLimit.map]
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  dsimp [plusMap]
  simp only [HasColimit.isoOfNatIso_ι_hom_assoc, ι_colimMap]
  dsimp [IsLimit.conePointUniqueUpToIso, HasLimit.isoOfNatIso, IsLimit.map]
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))
        (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
        (_ :
          ∀ (i : (Cover.index j.unop (F.obj k)).R),
            (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =
              (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) ≫
      colimit.ι (diagram J (F.obj k) X) j =
    (limit.lift (F ⋙ diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))).app j ≫
      colimit.ι (limit (F ⋙ diagramFunctor J D X)) j ≫
        (colimitLimitIso (F ⋙ diagramFunctor J D X)).hom ≫
          limit.lift (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
              ((Cones.postcompose
                    (NatIso.ofComponents fun k =>
                        colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).hom).obj
                (limit.cone (colimit (Functor.flip (F ⋙ diagramFunctor J D X))))) ≫
            limit.π (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X)) k

Next tactic:
rw [limit.lift_π]
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  dsimp [plusMap]
  simp only [HasColimit.isoOfNatIso_ι_hom_assoc, ι_colimMap]
  dsimp [IsLimit.conePointUniqueUpToIso, HasLimit.isoOfNatIso, IsLimit.map]
  rw [limit.lift_π]
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))
        (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
        (_ :
          ∀ (i : (Cover.index j.unop (F.obj k)).R),
            (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =
              (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) ≫
      colimit.ι (diagram J (F.obj k) X) j =
    (limit.lift (F ⋙ diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))).app j ≫
      colimit.ι (limit (F ⋙ diagramFunctor J D X)) j ≫
        (colimitLimitIso (F ⋙ diagramFunctor J D X)).hom ≫
          ((Cones.postcompose
                      (NatIso.ofComponents fun k =>
                          colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).hom).obj
                  (limit.cone (colimit (Functor.flip (F ⋙ diagramFunctor J D X))))).π.app
            k

Next tactic:
dsimp
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  dsimp [plusMap]
  simp only [HasColimit.isoOfNatIso_ι_hom_assoc, ι_colimMap]
  dsimp [IsLimit.conePointUniqueUpToIso, HasLimit.isoOfNatIso, IsLimit.map]
  rw [limit.lift_π]
  dsimp
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))
        (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
        (_ :
          ∀ (i : (Cover.index j.unop (F.obj k)).R),
            (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =
              (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) ≫
      colimit.ι (diagram J (F.obj k) X) j =
    (limit.lift (F ⋙ diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))).app j ≫
      colimit.ι (limit (F ⋙ diagramFunctor J D X)) j ≫
        (colimitLimitIso (F ⋙ diagramFunctor J D X)).hom ≫
          limit.π (colimit (Functor.flip (F ⋙ diagramFunctor J D X))) k ≫
            (colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).hom

Next tactic:
rw [ι_colimitLimitIso_limit_π_assoc]
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  dsimp [plusMap]
  simp only [HasColimit.isoOfNatIso_ι_hom_assoc, ι_colimMap]
  dsimp [IsLimit.conePointUniqueUpToIso, HasLimit.isoOfNatIso, IsLimit.map]
  rw [limit.lift_π]
  dsimp
  rw [ι_colimitLimitIso_limit_π_assoc]
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))
        (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
        (_ :
          ∀ (i : (Cover.index j.unop (F.obj k)).R),
            (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =
              (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) ≫
      colimit.ι (diagram J (F.obj k) X) j =
    (limit.lift (F ⋙ diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F))).app j ≫
      (limit.π (F ⋙ diagramFunctor J D X) k).app j ≫
        (colimit.ι (Functor.flip (F ⋙ diagramFunctor J D X)) j).app k ≫
          (colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).hom

Next tactic:
simp_rw [← Category.assoc, ← NatTrans.comp_app]
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
    [HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
    [ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
    (S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) (k) :
    liftToPlusObjLimitObj.{w, v, u} F X S ≫ (J.plusMap (limit.π F k)).app (op X) = S.π.app k := by
  dsimp only [liftToPlusObjLimitObj]
  rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]
  congr 1
  dsimp
  rw [Category.assoc, Category.assoc, ← Iso.eq_inv_comp, Iso.inv_comp_eq, Iso.inv_comp_eq]
  refine' colimit.hom_ext (fun j => _)
  dsimp [plusMap]
  simp only [HasColimit.isoOfNatIso_ι_hom_assoc, ι_colimMap]
  dsimp [IsLimit.conePointUniqueUpToIso, HasLimit.isoOfNatIso, IsLimit.map]
  rw [limit.lift_π]
  dsimp
  rw [ι_colimitLimitIso_limit_π_assoc]
  simp_rw [← Category.assoc, ← NatTrans.comp_app]
  

Proof state:
case e_a
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁹ : Category.{max v u, w} D
inst✝⁸ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝⁶ : ConcreteCategory D
inst✝⁵ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)
K : Type (max v u)
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : C
S : Cone (F ⋙ plusFunctor J D ⋙ (evaluation Cᵒᵖ D).obj (op X))
k : K
j : (Cover J (op X).unop)ᵒᵖ
⊢ Multiequalizer.lift (Cover.index j.unop (F.obj k)) (multiequalizer (Cover.index j.unop (limit F)))
        (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
        (_ :
          ∀ (i : (Cover.index j.unop (F.obj k)).R),
            (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.fstTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.fst (Cover.index j.unop (F.obj k)) i =
              (fun i => Multiequalizer.ι (Cover.index j.unop (limit F)) i ≫ (limit.π F k).app (op i.Y))
                  (MulticospanIndex.sndTo (Cover.index j.unop (F.obj k)) i) ≫
                MulticospanIndex.snd (Cover.index j.unop (F.obj k)) i) ≫
      colimit.ι (diagram J (F.obj k) X) j =
    ((limit.lift (F ⋙ diagramFunctor J D X) ((diagramFunctor J D X).mapCone (limit.cone F)) ≫
              limit.π (F ⋙ diagramFunctor J D X) k).app
          j ≫
        (colimit.ι (Functor.flip (F ⋙ diagramFunctor J D X)) j).app k) ≫
      (colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ diagramFunctor J D X)) k).hom

Next tactic:
rw [limit.lift_π, Category.assoc]