-- Module: Mathlib.CategoryTheory.Sites.LeftExact
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : Cᵒᵖ
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
m :
  S.pt ⟶
    Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
      (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫ NatTrans.app (Cone.π (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))) j =
      S.π.app j
preservesLimit :
  ∀ (k : K),
    CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X))
        (CategoryStruct.comp (Y :=
          limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
          (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
              (m ≫
                  Iso.hom
                    (HasColimit.isoOfNatIso
                      (IsLimit.conePointUniqueUpToIso
                        (isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F))
                        (limit.isLimit (F ⋙ J.diagramFunctor D (unop X))))) :
                S.pt ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
              (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D (unop X))) :
                colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))) ⟶
                  limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X))))) :
            S.pt ⟶ limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
          (Iso.inv
              (HasLimit.isoOfNatIso
                (Iso.symm
                  (NatIso.ofComponents
                    (fun (k : K) ↦
                      colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ J.diagramFunctor D (unop X))) k)
                    ⋯))) :
            limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))) ⟶
              limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)))
        (limit.π (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) k) =
      CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X))
        (limit.lift (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) S)
        (limit.π (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) k)
⊢ Eq (α := S.pt ⟶ limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X))
    (CategoryStruct.comp (Y :=
      limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
      (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
          (m ≫
              Iso.hom
                (HasColimit.isoOfNatIso
                  (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F))
                    (limit.isLimit (F ⋙ J.diagramFunctor D (unop X))))) :
            S.pt ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
          (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D (unop X))) :
            colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))) ⟶
              limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X))))) :
        S.pt ⟶ limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
      (Iso.inv
          (HasLimit.isoOfNatIso
            (Iso.symm
              (NatIso.ofComponents
                (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ J.diagramFunctor D (unop X))) k)
                ⋯))) :
        limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))) ⟶
          limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)))
    (limit.lift (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) S)

Tactic:
refine limit.hom_ext (fun k => ?_)

Premises used: CategoryTheory.Limits.limit.hom_ext