-- 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 s
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape.{s, s, t, w} 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 : C
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
k : K
j : (J.Cover (unop (op X)))ᵒᵖ
e_a :
  Eq (α := multiequalizer ((unop j).index (limit F)) ⟶ (J.plusObj (F.obj k)).obj (op X))
    (Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F)))
        (fun (x : (unop j).Arrow) ↦ Multiequalizer.ι ((unop j).index (limit F)) x ≫ NatTrans.app (limit.π F k) (op x.Y))
        ⋯ ≫
      colimit.ι (J.diagram (F.obj k) X) j)
    (CategoryStruct.comp (Y := (limit (F ⋙ J.diagramFunctor D X)).obj j)
      (NatTrans.app (limit.lift (F ⋙ J.diagramFunctor D X) (Functor.mapCone (J.diagramFunctor D X) (limit.cone F))) j)
      (CategoryStruct.comp (Y := colimit (limit (F ⋙ J.diagramFunctor D X)))
        (colimit.ι (limit (F ⋙ J.diagramFunctor D X)) j)
        (CategoryStruct.comp (Y := limit (colimit (F ⋙ J.diagramFunctor D X).flip))
          (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D X)) :
            colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ limit (colimit (F ⋙ J.diagramFunctor D X).flip))
          (NatTrans.app
              (Cone.π
                (Functor.obj
                  (Cone.postcompose
                    (Iso.hom
                      (NatIso.ofComponents
                        (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯)))
                  (limit.cone (colimit (F ⋙ J.diagramFunctor D X).flip))))
              k :
            Functor.obj.{s, t, s, w}
                (Functor.obj (Functor.const K)
                  (Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
                    (Functor.obj
                      (Cone.postcompose
                        (Iso.hom
                          (NatIso.ofComponents
                            (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯)))
                      (limit.cone (colimit (F ⋙ J.diagramFunctor D X).flip)))))
                k ⟶
              (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)).obj k))))
⊢ Eq (α := multiequalizer ((unop j).index (limit F)) ⟶ (J.plusObj (F.obj k)).obj (op X))
    (Multiequalizer.lift ((unop j).index (F.obj k)) (multiequalizer ((unop j).index (limit F)))
          (fun (x : (unop j).Arrow) ↦
            Multiequalizer.ι ((unop j).index (limit F)) x ≫ NatTrans.app (limit.π F k) (op x.Y))
          ⋯ ≫
        colimit.ι (J.diagram (F.obj k) X) j :
      multiequalizer ((unop j).index (limit F)) ⟶ colimit (J.diagram (F.obj k) X))
    (NatTrans.app (limit.lift (F ⋙ J.diagramFunctor D X) (Functor.mapCone (J.diagramFunctor D X) (limit.cone F))) j ≫
        CategoryStruct.comp (Y := colimit (limit (F ⋙ J.diagramFunctor D X)))
          (colimit.ι (limit (F ⋙ J.diagramFunctor D X)) j :
            (limit (F ⋙ J.diagramFunctor D X)).obj j ⟶ colimit (limit (F ⋙ J.diagramFunctor D X)))
          (CategoryStruct.comp (Y := limit (colimit (F ⋙ J.diagramFunctor D X).flip))
              (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D X)) :
                colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ limit (colimit (F ⋙ J.diagramFunctor D X).flip))
              (CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)))
                  (limit.lift (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
                    (Functor.obj
                      (Cone.postcompose
                        (Iso.hom
                          (NatIso.ofComponents
                            (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯)))
                      (limit.cone (colimit (F ⋙ J.diagramFunctor D X).flip))))
                  (limit.π (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) k) :
                limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ (J.plusObj (F.obj k)).obj (op X)) :
            colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ (J.plusObj (F.obj k)).obj (op X)) :
      multiequalizer ((unop j).index (limit F)) ⟶ (J.plusObj (F.obj k)).obj (op X))

Tactic:
rw [limit.lift_π]

Premises used: CategoryTheory.Limits.limit.lift_π